IC-NRLF 


LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


Class 


SECONDARY     STRESSES 


IN 


BRIDGE    TRUSSES 


BY 

C.    R.    GRIMM,    C.E. 

MEMBER    OF    THE    AMERICAN    SOCIETY    OF    CIVIL    ENGINEERS, 

MEMBER     OF     THE     AMERICAN     ASSOCIATION     FOR 

THE    ADVANCEMENT  OF    SCIENCE,    MEMBER 

OF   THE    AMERICAN    GEOGRAPHIC 

SOCIETY 


FIRST  EDITION 

FIRST    THOUSAND 


NEW   YORK 

JOHN    WILEY    &    SONS 
LONDON:    CHAPMAN    &   HALL,    LIMITED 

1908 


OF  THE 

UNIVERSITY   ) 

OF  / 


"TC-V 


COPYRIGHT,  1908, 

BY 
C.  R.  GRIMM 


Stanbope  fl>res& 

F.    H.   GILSON     COMPANY 
BOSTON.     U.S.A. 


PREFACE. 

THIS  book  owes  its  existence  to  Dr.  J.  A.  L.  Waddell,  Member 
of  the  American  Society  of  Civil  Engineers,  who  proposed  to  the 
writer  the  treatment  of  secondary  stresses  as  a  timely  undertaking. 

A  comprehensive  treatise  on  secondary  stresses  with  numerous 
numerical  examples  involves  an  extraordinary  amount  of  time 
and  labor  in  its  preparation,  and  considering  the  very  limited 
amount  of  time  at  the  writer's  disposal  for  such  an  attempt,  he 
thought  it  best  to  confine  himself  to  a  narrower  field,  as  otherwise 
the  publication  would  surely  have  been  unduly  delayed.  It  is 
owing  to  these  circumstances  that  we  discuss  principally  the  most 
important  secondary  stresses,  namely,  those  which  are  due  to 
riveted  joints  in  trusses. 

We  give  in  substance  four  principal  methods  of  calculation  of 
secondary  stresses,  which  the  reader  may  study  separately  and 
apply,  The  principle  of  least  work  leads  also  to  the  desired 
results,  but  it  is  far  removed  from  shortening  the  labor.  The 
nature  of  the  subject  forbids  a  very  quick  determination  of  the 
stresses,  and  definite  rules  or  formulas  cannot  be  given. 

The  problem  of  secondary  stresses  has  been  put  in  the  proper 
light  and  the  theories  worked  out  entirely  by  German  authors 
who  have  furnished  also  the  greatest  and  best  part  of  the  contri 
butions  to  the  literature  of  this  subject. 

In  preparing  these  notes  the  writer  consulted  a  great  many 
papers  and  books  most  of  which  are  noted  in  Chapter  XI.  He 
has  endeavored  to  be  clear  and  precise  in  his  statements,  but  if 
he  has  failed  in  this  he  begs  the  reader  to  be  indulgent  with  his 
shortcomings.  Should  he  succeed  in  interesting  his  readers  in 
this  new  field  for  American  bridge  literature,  and  in  rendering 
some  service  to  his  colleagues,  he  would  consider  himself  richly 
repaid  for  his  labors. 

C.  R.  G. 

GREATER  NEW  YORK,  1908. 


173028 


CONTENTS. 


CHAPTER    I. 

PAGE 

GENERAL  AND  HISTORICAL  NOTES 1-5 

Distinction  between  Primary,  Principal,  Additional  and  Secondary 

Stress    i 

Deformation  of  Trusses 2 

The  Most  Fruitful  Sources  of  Secondary  Stresses 2 

Displacement  of  Stresses 2 

Secondary  Stresses  as  Higher  Functions  of  the  Exterior  Loads      .    .  3 

Origin  of  the  Expression  "  Secondary  Stress  " 3 

Solution  of  the  Problem  by  Manderla 3 

Difficulties  of  the  Problem 4 

Further  Contributions  to  the  Subject  by  Engesser,  Winkler,  Lands- 
berg,  Miiller-Breslau,  Ritter  and  Mohr 4 

Cases  of  Examination  of  Secondary  Stresses 5 

CHAPTER    II. 

NATURE  OF  THE  PROBLEM  AND  MEANS  FOR  ITS  SOLUTION  .  .  6-19 

Deflection  of  a  Pin-Connected  and  a  Riveted  Truss 6 

Manderla's  Supposition 6 

Deformation  of  Riveted  Members,  Resembling  Usually  the  Letter  S  7 

The  Angle  of  Deflection 7 

The  Conditions  of  Equilibrium  for  a  Deformed  Bar 7 

Solution  Effected  by  Means  of  the  Equation  of  the  Elastic  Line  ...  8 
Simultaneous  Performance  of  the  Calculation  for  the  Entire  Truss 

Load 8 

Simplified  Assumptions  —  the  Method  of  Influence  Lines  ....  9 

Disturbing  Influences 9 

Determination  of  the  Angle-Alterations  A« 10 

Relations  between  the  Angle  of  Deflection  and  the  End  Moments  12 
Determination  of  the  Angle  Included  Between  Successive  Positions 

of  the  Axis  of  a  Bar,  Belonging  to  a  Truss  with  Frictionless  Pins, 

which  is  Acted  Upon  by  Exterior  Forces  16 

Work  Done  by  a  Couple if 

The  Elastic  Line  of  a  Straight  Beam  Represented  as  an  Equilibrium 

Curve  after  Mohr 17 

v 


vi  CONTENTS. 

CHAPTER   III. 

PAGE 
MANDERLA'S  METHOD 20-34 

Assumptions  Made  for  the  Solution  of  the  Problem  of  Secondary 

Stresses  Caused  by  Riveted  Joints 20 

Other  Elements  of  Influence   than   Riveted   Joints  on   Secondary 

Stresses 20 

Manderla's  Course  of  Investigation 20 

Differences  between  Compression-  and  Tension-Members     ....  21 

The  Deflection  Angles 24 

Compression-Members 25 

Tension-Members      27 

Determination   of   the    Deflection-Angles   from    the    Conditions   of 

Equilibrium 30 

Secondary  Stresses  Found  by  Trial -Computations 53 


CHAPTER    IV. 

MULLER-BRESLAU'S  METHOD      35-43 

Derivation  of  the  Fundamental  Equations , 35 

Assumed  Deformation  of  Triangles 35 

Values  of  the  Deflection  Angles      36 

Determination  of  the  Quantities  U  for  a  Truss 38 

Method  of  Influence  Lines 40 


CHAPTER    V. 

RITTER'S  METHOD        44-51 

Notation  of  Quantities      44 

Fundamental  Equations 46 

Graphic  Solution 46 

More  Exact  Method 49 


CHAPTER    VI. 

MOHR'S  METHOD 52-57 

Determination  of  the  Unknown  Quantities       52 

Determination  of  the  Angles  if/,  Pratt  Truss  as  Example      ....  53 

The  Bending  Moments  Expressed  as  Functions  of  the  Angles  <f>  and  ^  54 

Resulting  Equations  and  their  Solution 57 


CONTENTS.  vii 

CHAPTER    VII. 

PAGE 
METHOD  OF  LEAST  WORK 0    .    „    .    .         58-67 

A  Riveted  Triangle  as  an  Example 58 

The  Unknowns  of  the  Problem 59 

Influence  of  the  Sectional  Areas  on  the  Stresses 59 

Application  of  the  Principle  of  Least  Work 60 

Determination  of  the  Bending  Moments  and  the  Secondary  Stresses  63 

Displacement  of  the  Stresses 64 

Test  of  Accuracy  by  a  Check-Calculation 64 

CHAPTER    VIII. 

OTHER    CAUSES    OF    SECONDARY    STRESSES    THAN     RIVETED 

JOINTS  IN  MAIN  TRUSSES       68-84 

Eccentricities 68 

Loads  Between  Panel -Points  in  the  Plane  of  the  Truss 69 

Effects  of  Dead  Load 69 

Loads  Between  and  at  the  Panel-Points  of  a  Member  Supposed  to 

Turn  Freely  Around  a  Pin 69 

Bottom-Chord  Eyebar  as  an  Example       70 

Changes  in  Temperature 73 

Misfits 74 

Brackets  on  Posts      75 

Unsymmetrical  Connections 75 

Curved  Members 75 

Pin-Joints        75 

Greatest  Diameter  of  a  Pin,  if  an  Eyebar  Shall  Turn  Freely  ...  76 
Amount  of  Secondary  Stress  in  an  Eyebar  Due  to  Frictional  Resist- 
ance        77 

Friction  at  Supports      78 

Cross-frames 78 

Amounts  of  Secondary  Stresses  in  the  Suspenders  of  a  Two-Track 

Railway  Through  Bridge 79 

Amounts  of  Secondary  Stresses  in  the  Posts  of  a  Four-Track  Rail- 
way Through  Bridge 79 

Yielding  of  Foundations  and  Settlements  of  Masonry        80 

Trusses  Affected  by  a  Displacement  of  Their  Supports 80 

Continuous  Truss  or  Girder  Over  Three  Supports 81 

Two-Hinged  Arch 81 

Horizontal  Thrust  of  a  Two-Hinged  Arch  Caused  by  Vertical  Load- 
ing, Changes  in  Temperature  and  Horizontal  Yielding  of  the  Sup- 
ports    83 

Bridge  Across  the  Emperor  William  Canal  at  Griinenthal,  Germany, 

as  an  example 83 


viii  CONTENTS. 

CHAPTER    IX. 

PAGE 

IMPACT 85-9! 

The  Propagation  of  Stresses  in  Elastic  Bodies  and  the  Laws  of  Sound  86 
Ritter,  Mach,  Radinger  and  Steiner  on  the  Propagation  of  Impulses 

and  Vibrations 86 

Radinger  on  Bridge-Stresses  by  Fast  Running  Trains 87 

Zimmermann's  Rigid  Solution  of  the  Problem  of  Impact  in  the 

Simplest  Case 87 

Zimmermann's  Impact  Formula 90 

CHAPTER     X. 

EXAMPLES  AND  CONCLUDING  REMARKS 92-137 

Amounts  of  Secondary  Stresses  in  Three  Warren  Trusses  without 

Verticals      94-99 

Amounts  of  Secondary  Stresses  in  Two  Warren  Trusses  with  Verticals  100-104 

Discussion  of  Secondary  Stresses 105-106 

Amounts  of  Secondary  Stresses  in  Three  Pratt  Trusses 107-115 

Amounts  of  Secondary  Stresses  in  Two  Double  Intersection  Warren 

Trusses  without  Verticals 116-119 

Amounts  of  Secondary  Stresses  in  One  Double  Intersection  Warren 

Truss  with  Verticals 120-122 

Discussion  of  Secondary  Stresses  continued 123 

Amounts  of  Secondary  Stresses  in  a  Parabola  Truss       124-125 

Amounts  of  Secondary  Stresses  in  a  Continuous  Truss 126-127 

Discussion  of  Secondary  Stresses  continued 128 

Points  to  be  Observed  for  the  Reduction  of  Secondary  Stresses  in 

Trusses  Due  to  Static  Loads 128-130 

Secondary  Stresses  Due   to  Riveted   Connections   Between   Floor- 
beams  and  Main  Trusses 130-131 

Remarks  on  Floorbeams 132 

Points  to  be  Observed  for  the  Reduction  of  Impact  and  Vibrations  132-133 

Bridge  Collisions I33~I35 

Remarks  on  Calculations  and  their  Use I35~I37 

CHAPTER    XI. 

LITERATURE 138-140 

Concerning  Secondary  Stresses 138-140 

Concerning  Impact  and  Vibrations 140 


SECONDARY    STRESSES     IN 
BRIDGE     TRUSSES. 


CHAPTER    I. 
GENERAL   AND   HISTORICAL   NOTES. 

BEFORE  entering  upon  a  discussion  of  secondary  stresses,  a 
subject  which  has  been  very  obscure  up  to  about  twenty-five  years 
ago,  it  is  proper  to  show  to  the  reader  the  distinction  the  Germans 
make  between  different  stresses,  so  that  he  can  see  at  the  outset 
the  meaning  they  attribute  to  the  expression,  "Secondary  Stress." 

A  bridge  may  be  exposed  to  the  influence  of  a  number  of  causes, 
dead  load,  live  load,  wind  pressure,  centrifugal  and  braking 
forces,  changes  of  temperature,  yielding  of  masonry  (as,  for 
instance,  in  statically  indeterminate  arches,  continuous  beams 
and  trusses),  impact,  etc.  Any  one  of  these  causes  produces 
primary  stresses,  which  are  separated  into  two  classes.  Such 
that  are  due  entirely  to  dead  load  and  live  load  are  called  by 
German  writers  Hauptspannungen  (principal  stresses),  while 
those  due  to  any  other  cause  are  called  Zusatzspannungen 
(additional  stresses).  The  resultant  of  the  primary  stresses 
passes  through  the  center  of  gravity  of  the  section  and  acts 
along  the  axis  of  the  member,  producing  either  an  elongation 
or  a  shortening.  The  remainder  of  the  stresses  are  bending, 
shearing  and  torsional  stresses;  they  bend,  displace  and  twist, 
and  are  comprised  under  the  expression,  secondary  stresses 
(German  —  Nebenspannungen,  Sekunddrspannungen;  French  - 
efforts  secondaires). 


2  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

In  other  words,  in  a  truss  with  frictionless  pins,  representing  an 
ideal  truss,  the  axes  of  the  truss  members  remain  straight  during 
deformation,  while  in  a  riveted  truss  these  axes  are  subjected  to 
deformations  accompanied  by  secondary  stresses. 

In  the  computations  of  bridge  stresses  we  always  proceed  under 
the  supposition  that  the  joints  are  provided  with  frictionless  pins; 
a  supposition,  which  is  either  not  at  all  realized  in  the  structure, 
or  at  best  in  an  imperfect  manner,  so  that  each  truss  member 
is  bound  to  be  deformed,  that  is  to  say,  each  member  during  the 
action  of  the  load  is  subjected  to  bending  moments  in  the  plane 
of  the  truss,  and  its  axis  cannot  remain  straight.  Not  only  the 
members  composing  a  riveted  truss,  but  all  those  that  go  into  the 
riveted  cross  frames  and  the  bracings  are  subjected  to  secondary 
stresses.  The  riveted  joints  of  the  main  trusses  in  particular, 
as  also  the  stiff  connections  between  floor  system  and  trusses  form 
the  most  fruitful  source  of  secondary  stresses.  For  this  reason, 
we  will  give  for  the  present  some  general  remarks  about  the  second- 
ary stresses  in  riveted  trusses  and  discuss  later  those  stresses  which 
arise  from  the  riveting  between  main  trusses  and  floorbeams,  as 
also  several  other  groups  of  secondary  stresses. 

The  deflection  of  an  ideal  truss  with  frictionless  pins,  due  to 
external  forces,  causes  alterations  of  leverarms,  which  the  computer 
of  primary  stresses  completely  ignores,  and  rightfully  so,  since 
these  alterations  have  no  practical  significance  whatever.  The 
original  leverarms  are  very  great  in  comparison  with  the  altera- 
tions that  have  taken  place  after  deflection  and,  therefore,  these 
changes  are  neglected.  The  lines  of  stresses  of  the  different  mem- 
bers around  a  panelpoint  still  pass  through  a  common  center. 
On  the  other  hand,  when  we  attempt  the  computation  of  secondary 
stresses  in  a  riveted  truss,  we  have  to  deal  with  small  leverarms, 
because  each  member  of  the  truss  has  been  bent  and  the  altera- 
tions of  these  leverarms  cannot  be  ignored  without  some  sufficient 
reason.  The  resultant  stress  in  each  bar  does  not  act  any  longer 
parallel  to  the  axis  of  the  bar,  on  the  contrary,  it  forms  an  angle 
with  it,  which  makes  the  leverarm  and  the  lines  of  stresses  of  the 


GENERAL    AND    HISTORICAL    NOTES  3 

different  members  around  a  panelpoint  pass  no  longer  through 
one  and  the  same  mathematical  point,  that  is,  through  a  panelpoint 
of  the  truss. 

By  taking  into  account  the  influence  of  the  deformations  on  the 
alterations  of  leverarms,  the  secondary  stresses  appear  as  higher 
functions  of  the  exterior  forces,  while  in  neglecting  these  influences 
they  appear  as  linear  functions  of  the  exterior  forces,  and  one  of  the 
difficulties  is  removed. 

It  is  natural  that  the  deformation  of  a  compression  member  plays 
a  greater  role  than  that  of  a  tension  member,  but  if  ample  provision 
against  buckling  is  made  in  the  design  of  compression  members, 
the  neglect  of  the  deformations  in  the  calculations  is  justified. 

The  expression,  Sekunddrspannung  (secondary  stress),  origi- 
nated with  Professor  Asimont  of  the  polytechnic  school  in  Munich, 
Bavaria.  In  a  paper,  "  Hauptspannung  und  Sekundarspannung" 
(primary  and  secondary  stress),  which  was  published  in  1880  in 
Zeitschrijt  jur  Baukunde,  Asimont  discusses  the  effects  of  eccentric 
loading  on  a  column,  and  here  he  made  the  distinction  between 
primary  or  direct  stress  and  that  due  to  a  couple,  producing  a 
bending  stress,  which  he  called  Sekundarspannung.  But  long 
before  Asimont  coined  that  expression,  engineers  considered 
secondary  stresses  in  their  designs. 

The  subject  of  secondary  stresses  being  one  of  importance,  the 
polytechnic  school  in  Munich,  in  the  year  1877,  offered  a  prize  on 
the  solution  of  the  problem  of  how  to  calculate  these  stresses  in 
riveted  trusses.  In  formulating  the  problem,  Asimont  suggested 
that,  owing  to  the  fact  that  the  lines  of  the  resulting  stresses  no 
longer  pass  through  the  centers  of  the  paneljoints,  its  solution 
might  be  effected  by  the  employment  of  Euler's  equation  of  the 
elastic  line.  The  prize  was  awarded  to  Manderla,  in  1879,  whose 
excellent  solution  is  found  in  a  highly  scientific  and  mathematical 
paper,  published  in  1880,  in  Allgemeine  Bauzeitung,  under  the  title, 
"Die  Berechnung  der  Sekundarspannungen,  welche  im  einfachen 
Fachwerke  infolge  starrer  Knotenverbindungen  enstehen"  (The 
calculation  of  secondary  stresses  which  occur  in  simple  trusses  as  a 


4  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

consequence  of  rigid  joints).  The  nature  of  the  problem  of 
secondary  stresses  in  riveted  trusses  is  such  that  it  offered  great 
obstacles  to  a  solution;  in  fact,  this  problem  is  one  of  the  most 
difficult  in  technical  mechanics,  and  although  Manderla's  solution 
is  a  very  great  step  forward,  the  problem  in  all  its  aspects  has  not 
yet  been  completely  solved. 

Before  the  appearance  of  Manderla's  solution,  Engesser  had 
published  an  approximate  method.  In  the  year  1881,  the  late 
Professor  Winkler  gave  a  lecture  on  secondary  stresses  before  an 
organization  of  engineers  and  architects  in  Berlin,  in  which  he 
said  that  for  some  years  past  he  had  paid  attention  to  the  subject. 
This  lecture  is  published  in  Deutsche  Bauzeitung  in  1881,  under 
the  title,  "Die  Sekundarspannungen  in  Eisenkonstruktionen  " 
(Secondary  stresses  in  iron  constructions).*  In  1885  Professor 
Landsberg  contributed  a  graphical  solution  under  the  assumption 
that  the  chords  alone  are  riveted;  and  in  1886  Professor  Miiller- 
Breslau  made  an  analytical  contribution.  Professor  Ritter,  in 
1890,  gave  a  graphical  solution,  and  in  the  years  1892  and  1893 
Professor  Engesser  published  a  book  on  secondary  and  additional 
stresses,  the  latter  expression  to  be  taken  in  the  sense  already 
explained.  In  this  book  a  systematic  representation  of  the  whole 
subject  is  given;  the  treatment  is  analytical  throughout.  Pro- 
fessor Mohr  contributed  in  1892  an  analytical  method  in  "  Der 
Civil  Ingenieur,"  which  can  also  be  found  in  his  very  valuable 
book,  "  Abhandlungen  aud  dem  Gebiete  der  technischen 
Mechanik,"  published  in  1906,  in  which  he  treats  in  a  masterly 
manner  subjects  appertaining  to  technical  mechanics. 

The  fundamental  truths  and  the  methods  of  calculations  that 
came  to  light  in  studying  the  problem  that  is  here  under  consider- 
ation, we  owe  to  German  scientists.  The  results  they  obtained 
were  soon  made  use  of  by  German  bridge  constructors,  as  is 
noticeable  in  their  designs,  but  this  is  a  point  to  which  we  will 
return  at  another  place. 

*  Very  extensive  investigations  can  be  found  in  Winkler's  "  Theorie  der 
Briicken,"  II  Teil. 


GENERAL    AND    HISTORICAL    NOTES  5 

The  calculations  of  secondary  stresses  are  very  extensive  and 
require  therefore  much  time;  but  under  simplified  assumptions, 
and  with  the  requirement  to  find  these  stresses  for  only  one  given 
load  system,  the  calculations  can  be  performed  with  comparative 
speed.  It  is  hardly  in  the  nature  of  the  problem  to  give  empirical 
rules  and  formulas.  Although  in  common  cases  there  is  no 
necessity  for  such  calculations,  yet  in  particular  cases  secondary 
stresses  should  be  investigated;  for  instance,  in  cases  where  we 
can  expect  them  to  be  of  great  magnitude,  or  where  a  bridge  has  to 
carry  much  greater  loads  than  those  for  which  it  has  been  designed, 
which  is  true  of  old  bridges.* 

The  margin  of  safety  provided  for  in  our  specifications  must 
cover  the  secondary  stresses  without  impairing  the  safety  of  the 
structure. 

The  nature  of  a  truss,  its  details,  as  well  as  the  dimensions  of 
its  individual  members,  are  of  great  importance  for  the  reduction 
of  secondary  stresses,  but  in  order  to  obtain  the  best  results  even 
attention  must  be  paid  to  the  manufacture,  erection  and  main- 
tenance of  bridges. 

*  W.  J.  Watson,  Concerning  the  investigation  of  Overloaded  Bridges,  Pro- 
ceedings Am.  Soc.  of  C.  E.,  April,  1906. 


CHAPTER    II. 
NATURE   OF   THE   PROBLEM  AND   MEANS   FOR  ITS  SOLUTION. 

IN  the  following  discussion  it  is  assumed  that  all  exterior  forces, 
to  whose  influence  a  truss  is  exposed,  are  acting  in  the  plane  of 
the  truss.  The  deflection  of  a  truss  with  supposed  frictionless 
pins  is,  strictly  speaking,  not  the  same  as  that  of  a  riveted  truss, 
all  other  conditions  being  the  same  for  both  trusses.  While  in 
the  former  case  the  problem  is  geometrical,  since  the  bars  can  turn 
freely  around  the  pins  and  remain  straight,  in  a  riveted  truss 
the  axes  of  the  bars  become  deformed  under  the  action  of  the 
load,  a  fact  which  must  be  considered,  together  with  the  moments 
of  inertia  of  the  sections,  in  determining  the  deflection.  But  the 
difference  in  the  deflection  between  these  two  cases  is  so  small 
that  its  consideration  has  no  practical  value. 

Manderla's  solution  is  based  on  the  supposition  that  the  posi- 
tions of  the  panelpoints  in  a  riveted  truss  under  the  action  of 
outer  forces  are  the  same  as  if  the  truss  were  provided  with  fric- 
tionless pins. 

In  Fig.  i,  representing  a  fragment  of  a  truss,  the  angle  formed 
by  the  two  bars  01  and  02  before  any  deformation  takes  place  is 
4-  102  =  a.  Under  the  supposition  of  frictionless  pins  this 
angle  a  will  be  changed  as  soon  as  the  truss  is  subjected  to 
the  influence  of  exterior  forces.  Let  this  change  be  &a,  so 
that  the  angle  included  between  the  two  bars  01  and  02  after 
deformation  equals  a  +  Aa. 

If  we  now  conceive  the  truss  to  have  joints  riveted  in  such  a 
manner  as  to  keep  the  ends  of  the  bars  absolutely  fixed,  then 
under  the  influence  of  the  outer  forces  the  originally  straight  bars 
will  be  deformed  and  the  angle  included  between  the  two  end 

6 


THE  PROBLEM  AND  ITS  SOLUTION         7 

tangents  o7\  and  oT2,  drawn  to  the  elastic  lines,  which  are  the 
deformed  axes  of  the  bars,  will  remain  unchanged  during  deform- 
ation, that  is  to  say,  the  angle  7\or2  =  a  or  =  the  original  angle 
before  any  deformation  took  place. 

Each  bar  will  be  subjected  to  bending  moments,  and  its  deform- 
ation generally,  but  not  always,  resembles  the  letter  S.     The  bars 


Fig.  i. 


must  be  considered  fixed  at  the  ends  and  under  the  influence  of 
an  axial  load,  while  the  effect  of  the  bending  moments  can  be 
conceived  as  consisting  in  the  reduction  of  the  changed  angle 
a  +  \a  to  its  original  magnitude  a. 

The  angle  r,  included  between  a  chord  of  a  deformed  bar  and 
its  corresponding  end  tangent,  as,  for  instance,  o/  and  or,  we 
will  call  the  angle  of  deflection. 

We  will  now  consider  a  single  bar  all  by  itself,  for  instance,  bar 
01  (see  Fig.  2),  by  passing  sections  close  to  the  panelpoints  and 
applying  the  resultant  P0  of  all  the  interior  forces,  which  inter- 
sects the  chord  of  the  deformed  bar  under  the  angle  &».  This 
resultant  is  resolved  into  the  force  50,  acting  along  the  axis  of  the 
deformed  bar,  the  transverse  force  Q0  and  the  moment  M0. 

The  equilibrium  requires  that 

-S0  =  s,,  G.  =  G,,  ^~-1  -  G,  =  o. 

M  +  M 

The  magnitude  of   the  transverse  force  Q  =  — is  princi- 


8 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


pally  dependent  on  the  signs  of  the  two  moments;  the  greater 
values  of  Q  correspond  to  equal  signs  of  M  and  consequently 
to  a  double  curvature  of  the  bar,  while  opposite  signs  of  the 
moments  mean  smaller  values  of  Q  and  a  single  curvature. 
The  transverse  force  Q  is  constant  for  the  entire  length  of  the  bar, 
as  its  ends  are  the  only  points  of  application  of  the  outer  forces, 
and  the  bending  moment  for  any  point  between  the  ends  of  the  bar 
is  variable  under  the  influence  of  Q  and  S. 

I  Manderla's  excellent  original  solution  considers  all  of  the  forces 
represented  in  Fig.  2.  He  proceeds  from  the  equation  of  the 
elastic  line,  employs  for  the  integrations  hyperbolic  functions,  and 
after  a  relation  between  the  bending  moments  and  the  angle  of 
deflection  ~  is  established,  expressing  the  latter  in  terms  of  a,  he 
obtains  the  desired  result  by  trial. 

Other  scientists,  who  have  occupied  themselves  with  a  solution 
of  this  problem,  proceed  from  the  assumption  that  the  influence 
of  the  deformation  on  the  leverarm  y  is  a  negligible  quantity,  or 
in  other  words,  for  well  designed  compression  members  — tension 
members  are  not  so  important  in  this  respect  —  the  leverarm  y, 
and  consequently  also  the  moment  Sy,  is  so  small  that  it  does  not 
need  to  be  considered.  The  transverse  forces  Q,  too,  are  in  most 
cases  small  enough  to  be  neglected.  These  suppositions  lead  to 
simplified  calculations. 

In  Manderla's  solution  the  secondary  stresses  appear  as  higher 


Fig  2. 


functions  of  the  exterior  forces,  which  means  that  the  calculations 
must  be  performed  simultaneously  for  the  entire  load  system. 


THE    PROBLEM    AND    ITS    SOLUTION  9 

If,  for  instance,  the  secondary  stresses  in  a  railroad  truss  have 
been  determined  for  a  certain  position  of  a  train,  we  have  no 
means  of  knowing  what  the  magnitude  of  these  secondary  stresses 
amount  to  as  soon  as  the  train  is  moved  to  some  other  position. 
The  method  of  influence  lines  does  not  hold  true  in  this  case,  while 
under  the  simplified  assumption,  that  is,  if  the  effect  of  the  moment 
Sy  on  the  final  result  is  neglected,  the  secondary  stresses  appear  as 
linear  functions  of  the  exterior  forces  and  the  investigator  may 
employ  the  method  of  influence  lines  if  he  so  desires.  The  fact 
that  secondary  stresses  do  not  increase,  or  decrease  in  direct  pro- 
portion to  the  exterior  loads  makes  the  so-called  "  factor  of  safety''* 
appear  as  the  "factor  of  uncertainty,"  and  uncertainties  must  be 
covered  by  the  margin  of  safety  in  our  specifications. 

The  assumptions  which  we  have  made  are  never  strictly 
realized  in  a  structure,  but  this  is,  of  course,  equally  true  for  the 
calculations  of  other  bridge  stresses.  It  is  within  the  range  of 
possibilities  that  the  deformation  of  some  truss  members  will  take 
place  outside  of  the  plane  of  the  truss  and  they  may  even  be  sub- 
jected to  the  influence  of  torsional  moments.  It  must  also  be 
borne  in  mind  that  the  desire,  of  the  bridge  engineer  to  save  some 
weight  often  results  in  sizes  of  gusset  plates  so  small  that  the  state  of 
fixity  of  the  bars  is  complied  with  in  a  very  imperfect  manner. 
A  variation  in  the  value  of  the  modulus  of  elasticity  and  possible 
erection  stresses  play  also  a  role  in  making  the  results  somewhat 
uncertain. 

In  the  first  place  we  will  deduce  some  fundamental  conclusions 
from  technical  mechanics,  which  are  indispensable  for  a  clear 
understanding  of  the  different  methods  of  calculations  of  second- 
ary stresses. 

*  (i)  "Factors  of  safety,"  Eng.  News,  Sept.  6,  1906. 

(2)  "The  Investigation  of  Old  Bridges  a  Phase  of  Maintenance  Engineering," 
Eng.  News,  Sept.  13,  1906. 


10  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

i.   Determination  of  the  Angle  Alterations  Aa. 

An  important  problem  in  the  theory  of  trusses,  forming  the 
foundation  of  Manderla's  solution,  consists  in  the  determination 
of  the  alterations  Aa  of  the  angles  a  between  any  two  adjacent 
pin-connected  bars  of  a  truss  when  the  latter  is  acted  upon  by 
outer  forces.  Aa  will  be  expressed  in  terms  of  the  primary 
stresses,  which  are  known  quantities. 

The  deformation  of  the  angle  a,  in  the  triangle  ABC  of  Fig.  3, 
is  due  to  the  elastic  changes  of  all  three  sides,  and  this  deformation 
is  obtained  by  determining  the  influence  of  each  bar  on  the 
deformation  successively,  whereupon  the  results  are  summed  up. 
This  is  done  in  assuming  that  each  bar  in  turn  is  elastic,  the  other 
two  being  non-elastic.  If  the  side  AC  is  alone  elastic,  experienc- 
ing a  contraction  to  the  amount  of  AL3,  it  will  be  forced  to  revolve 
around  the  apex  A ,  while  the  side  BC  revolves  around  B,  the  side 
AB  being  supposed  to  be  held  fast. 

From  the  figure  we  have 

Aa13  =  al  —  /?!, 

where  the  index  i  refers  to  the  angle  and  the  index  3  to  the  side. 
Further: 

«!  ==  180°  -  a,  -a3 
and 

/?!  =  180°  -  a,  -  a3  -  e  +  7 

consequently : 

Aals  =    -  7  +  s. 

The  angle  7  is  found  from  the  equation  BCy  X  cos  al  =--    a  or 

a  a      AZ,3 

7  =:   „„  =  T  =   — ; —  on   account   of   the   similarity   of  the 

BC  cos  ai       b         h 

triangles  BCF  and  CDE.      We  have  also  r  X  —  =  ~T->  and 

o       L3         L3h 

a  &AZ, 

since  the  angle   e  ==  — ,  we  find  by  substitution:    e  =  •  . 

^3  LJl 


THE    PROBLEM    AND    ITS    SOLUTION  II 

With  these  values  for  7  and  e  the  deformation  becomes 

A«13  =        — — ^  +  3  • 

If  53  denotes  the  total  stress  in  bar  AC  and  A3  its  cross-section, 

then  -~  =  s3,  which  is  the  stress  per  unit  of  area.       Calling  E 

-4  3 
the  modulus  of  elasticity,  the   deformation  of  the  angle  alt  due 


only  to  the  alteration  in  the  length  of  the  bar  AC,  is  expressed  by 
53        L3  -  b  _      _Sy_  ^ 

1 3  TT*  A  i  77*  A  2  T-I  2  * 

&A3  tl  £LA3  JtL 

If  the  bar  BC  alone  is  elastic,  we  find  in  a  similar  manner, 
A«12  =  —  -J?  cot  ci'3, 

and  under  the  supposition  that  AB  alone  is  subjected  to  a  change 
in  its  length,  we  have : 

AT  0*7"  C1       (   f  }  ( 

— ~ =  ~^~r~  =  ^rr -  ]  -  +  -  [  =  IT  }  cot  « 

r         Eyltr       EA<  r  r       r\        El 


Aa, 


cot 


12  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

The  total  deformation  of  angle  a  t  is  now  found  by  summing  up 
the  three  results,  just  found,  so  that  we  can  write: 

Ao^  =  Aan  +  Aa12  +  Aa13. 

In  substituting  the  values  in  this  equation  and  treating  the 
deformations  of  the  angles  a2  and  «3  in  like  manner,  we  finally 
arrive  at  the  three  equations: 


(i) 


If  we  attribute  to  a  tensile  stress  the  positive  and  to  a  com- 
pressive  stress  the  negative  sign,  we  can  obtain,  for  instance,  the 
positive  maximum  deformation  of  angle  a1  by  supposing  s1  a 
tension  and  s2  and  s3  a  compression,  while  the  negative  maximum 
deformation  of  the  same  angle  results  from  a  negative  ^  and  a 
positive  s2  and  sy 

2.   Relations  Between  the  Angle  of  Deflection  and  the  End 
Moments. 

The  Italian  engineer  Alberto  Castigliano  in  his  book,  "The'orie 
de  Pequilibre  des  Systemes  elastiques,"  demonstrated  and  intro- 
duced the  principle  of  the  derivative  of  work  and  the  principle 
of  least  work.*  The  first  quoted  principle  means,  in  our  case, 
that  if  we  express  the  work  of  deformation  of  a  bar  as  a  function 
of  the  outer  forces,  its  first  derivative  with  respect  to  a  moment 
equals  the  angle  of  revolution  of  the  bar.  If  the  first  derivative 
is  taken  with  respect  to  a  force,  we  obtain  the  displacement  in  the 
direction  of  the  force  of  its  point  of  application. 

In  Fig.  4,  let  AB  represent  a  beam  fixed  at  B  and  under  the 
influence  at  its  end  A  of  a  vertical  force  Q  and  a  moment  MQ. 

*  W.  Cain  :  Determination  of  the  Stresses  in  Elastic  Systems  by  the  Method 
of  Least  Work.  Transactions  Am.  Soc.  C.  E.,  Vol.  XXIV,  1891. 


THE  PROBLEM  AND  ITS  SOLUTION 


The  angle  r  included  between  a  horizontal  line  and  the  end  tan- 
gent drawn  to  the  elastic  line  of  the  beam  is  the  angle  sought, 


._£-. 


Mo 


Fig.  4. 


measured  by  the  length  of  a  circular  arc  (of  unit  radius)  and 
expressed  by  the  equation: 

T   M2    , 

-  dx  > 

dM 


o    2  El 


r  = 


r  M_ 

Jo  EI 


dx. 


EIdM0 

M  =  Moment  around  any  point  of  the  axis  of  the  beam, 
E  =  Modulus  of  elasticity, 
/.  =  Moment  of  inertia. 
We  will  now  consider  a  bar  of  double  curvature,  Fig.  5,  but  with 


Fig.  5- 


Fig.  6. 

the  express  understanding  that  the  end  moments  only  will  be 
considered  in  determining  the  deflection  angles  T. 


14 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


In  Fig.  6,  the  moments  are  represented  graphically,  M0  =  oa 
and  M1  =  i  b,  and  the  moment  area  appears  as  the  difference  of 
two  triangles,  so  that  we  can  write: 


Area  oiba  =  Aoafr  —  Aoi5. 
The  moment  in  reference  to  any  point  of  the  axis  of  the  bar  or 


and 


M  =    ~{M0  +  M,}  y  +  M0 


6M       __* 
dM0          I 


Applying  now  the  principle  of  the  derivative  of  work  for  the 
determination  of  the  angle  T  as  expressed  in  the  above  equation, 
we  can  write: 


,  _ 

~o  EIdM0 


which  gives: 

T'  =  JTr  i 
El  ( 


-  + 
3 


,  -  -  MQ  -  -  M0  - 
3  2  2 


-  + 
2 


or 


T    = 


{2  MQ  -  M,} 


(2) 


TJ  is  found  by  exchanging  M0  for  Mr 

The  deformation  shows  a  single  curvature  and  with  no  point 
of  contraflexure  when  a  bar  is  acted  upon  by  two  end  moments 
of  opposite  sign,  which  case  will  hereafter  be  exclusively  con- 
sidered, Figs.  7  and  8. 


THE    PROBLEM    AND    ITS    SOLUTION  15 

In  regard  to  Fig.  8,  M0  =  oa  and  M^=  i  b,  and  the  moment  area 
appears  as  a  trapezoid.  The  moment  at  any  point  of  the  axis  of 
the  bar  is  equal  to 


and 


M=  {M,  -M0(7  +  M0 
l 


dM  x 


Fig.  7 


consequently  we  have: 


C1M. 

T°  ~  Jo  E 


dM 
EIdM0 


dx  = 


or 


T    = 


and  also 


(3) 


16  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

3.  Determination  of  the  Angle  Included  Between  Two  Successive 
Positions  of  the  Axis  of  a  Bar,  Belonging  to  a  Truss  with 
Frictionless  Pins,  which  is  Acted  Upon  by  Exterior  Forces. 

We  assume  OA  and  OAl  are  the  directions  of  the  axis  of  the 
bar  "before"  and  "  after"  deformation,  so  that  the  angle  <!> 
included  between  OA  and  OAi  is  the  angle  sought;  see  Fig.  9. 
The  equation 


states  that  the  work  done  by  the  exterior  forces  equals  the  work 
done  by  the  interior  forces  in  case  of  equilibrium.  In  this  equa- 
tion P  designates  an  outer  force  and  S  its 
corresponding  stress,  while  A/  is  an  alteration  of 
the  length  of  a  bar  and  d  a  displacement  in  the 
direction  of  the  force  P  of  its  point  of  application. 
The  work  done  by  the  reactive  forces  is  not 
considered,  as  we  assume  that  the  points  of  sup- 
port are  immovable  in  the  direction  of  these 
forces.  The  above  equation  holds  true  for  any 
possible  displacements  §  and  alterations  A/  and 
for  any  values  of  P  which  are  independent  of 
the  d  and  A/. 

If  we  wish  to  calculate  any  particular  dr  in  the  direction  of  Pr, 
which  is  caused  by  a  given  load  system,  it  is  only  necessary  to  let 
all  of  the  forces  P  vanish,  except  Pr,  which  is  put  equal  to  unity. 
In  doing  this  the  new  equation  results: 


in  which  the  stresses  5  are  caused  by  the  force  unity  and  the 
A/  are  caused  by  the  given  exterior  forces,  §r  corresponding  to  the 
A/.  The  force  unity  may  represent  a  single  force  or  a  moment, 
in  which  latter  case  the  angle  of  revolution  ^',  measured  by  the 
length  of  the  circular  arc,  must  be  substituted  for  the  displace- 
ment dr  and  our  equation  becomes: 

(4) 


THE  PROBLEM  AND  ITS  SOLUTION        17 

We  will  now  show  that  the  expression  M  X  4>  represents  work 
as  it  should  do. 

In  Fig.  10,  OA  represents  a  bar,  having  its  center  of  revolution 
at  O  and  acted  upon  by  a  moment  M  =  Q  X  a.  The  angle  (p  is 


L= — &__      — 4«— a 

Fig.  10. 

measured  downward  from  OA   in  agreement  with  the  direction 
of  revolution  of  the  moment  Q  X  a,  which  is  the  same  as  that  of 
the  hands  of  a  clock. 
From  the  figure  we  have: 

Work  =  Q  X   {a  +  b\  $  -  Q  X  b</> 
=  Q  x  a^  =  M  X  0. 

4.   The  Elastic  Line  of  a  Straight  Beam  Represented  as  an 
Equilibrium  Curve  after  Mohr.* 

This  problem  was  first  solved  by  Mohr  nearly  forty  years  ago 


dx  dx 


Fig.  n. 


by  comparing  the  equation  of  the  elastic  line  with  that  of  an 
equilibrium  curve. 

*  Mohr:  "  Abhandlungen  aus  dem  Gebiete  der  technischen  Mechanik,"  1906. 


1 8  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

In  Fig.  n,  the  beam  is  subjected  to  a  vertical  load,  which  is 
continuous  but  non-uniform.  The  hatched  area  is  the  load  area 
of  the  beam  and  p  designates  the  variable  load  per  unit  of  length 
of  the  horizontal  projection  of  the  curve. 

Figure   12   is  a  fragment  of    the   force    polygon  and   Fig.  13 


Fig.  12. 

shows  the  infinitely  small  sides  mn  and  no  of  the  equilibrium 
polygon   (curve)  with  horizontal  and  vertical  projections  in  an 


_^  *'  —  ~ 

*--              dx  —           —  s» 

\d 


pdx 
Fig.  13. 

exaggerated  scale,  p  doc  being  the  load  at  point  n.     As  the  hatched 
triangles  are  similar  in  Figs.  12  and  13,  we  can  write: 

^2  -  P-fa 
dx       H 

Dividing  by  dx  we  have  the  differential  equation  of  the  equi- 
librium curve  for  vertical  forces,  that  is, 


dx2      H 


THE    PROBLEM    AND    ITS    SOLUTION  19 

The  differential  equation  of  the  elastic  line 

d?y  =  M_ 
doc2  ~  El 

is  of  the  same  form,  where  M  =  any  moment,  E  =  modulus  of 
elasticity  and  7  =  moment  of  inertia.     Consequently,  if  we  put 

M  =  p  and  El  =  H, 
or  '—  =  p  and  E    =  H , 

or  —j  -  p  and      i  «  H, 

we  see  that  the  elastic  line  can  be  conceived  as  an  equilibrium 
curve. 


CHAPTER    III. 
MANDERLA'S   METHOD. 

THE  problem  of  secondary  stresses  is  solved  under  the  assump- 
tion that  the  deformations  of  a  riveted  truss,  caused  by  the  exterior 
loading,  are  accomplished  in  the  plane  of  the  truss  and  that  no 
torsion  exists.  It  is  further  assumed  that  every  load  is  applied 
at  the  panelpoints  of  the  truss,  which  is  composed  of  members 
having  uniform  sections.  This  last  assumption  is  not  quite  true, 
since  the  truss  members  are  connected  together  by  means  of  gusset 
plates,  which  make  a  sudden  change  in  the  sectional  areas  at  the 
ends  of  the  members. 

There  are  other  elements  which  are  of  influence  on  the  stresses, 
but  which  can  be  examined  separately.  One-sided  connections 
hardly  need  any  consideration,  as  they  are  condemned  by  our 
specifications ;  but  the  effect  of  details  not  centrally  designed,  as 
also  the  effect  of  loads,  applied  at  any  point  between  the  panel- 
points,  as  dead  and  live  load,  wind  pressure  and  centrifugal  force, 
and  the  influence  of  a  change  in  temperature  on  the  secondary 
stresses,  may  be  calculated.  Of  considerable  importance  is  the 
effect  of  the  deformations  of  floorbeams  riveted  to  the  trusses  on 
the  stresses  in  truss  members,  but  this  is  a  problem  whose  analytical 
discussion  is  entirely  outside  the  range  of  our  considerations. 

The  progress  of  the  investigation  is  as  follows:  Proceeding 
from  the  supposition  that  the  positions  of  the  panelpoints  of  a 
riveted  truss  under  the  influence  of  loads  are  the  same  as  if  the 
truss  had  frictionless  pins,  the  alterations  of  all  the  angles  in  the 
triangles,  composing  the  truss,  are  calculated  according  to  equa- 
tions (i)  in  Chapter  II,  for  that  system  of  loads  for  which  we 
intend  to  determine  the  secondary  stresses.  But,  as  the  ends  of 
our  truss  members  are  supposed  to  be  rigidly  fixed,  any  alterations 


MANDERLA'S    METHOD  21 

in  the  angles  of  the  triangles  are  impossible,  in  consequence  of 
which  the  truss  members  become  deformed  under  the  action  of  the 
exterior  loads  and  are  subjected  to  bending  moments,  which  must 
reduce  the  supposed  changed  angles  to  their  original  magnitudes. 
The  end  bending  moments  are  now  introduced  into  the  calcu- 
lations as  the  unknowns,  and  relations  are  determined  between 
them  and  the  angles  of  deflection,  which  latter  angles  are  included 
between  the  chords  of  the  deformed  bars  and  the  end  tangents 
drawn  to  the  elastic  lines  of  the  bars.  Hereupon  the  angles  of 
deflection  are  expressed  in  terms  of  the  alterations  of  the  angles 
in  the  triangles  calculated  by  means  of  equations  (i)  for  the  desired 
load  system  under  the  assumption  of  frictionless  pins.  After  the 
unknown  bending  moments  have  been  determined,  it  is  easy  to 
find  the  secondary  stresses  according  to  known  formulas.  Man- 
derla  finds  the  desired  stresses  not  directly,  but  by  a  few  trial 
computations,  which  are  easily  performed. 

Before  we  proceed  with  the  analysis  of  the  stresses,  it  will  be 
necessary  to  give  some  remarks  on  the  action  of  the  forces  to 
which  the  truss  members  are  subjected,  and  to  point  out  the  differ- 
ences which  exist  between  the  compression  and  tension  members, 
as  shown  in  Figs.  14,  15,  16,  17,  18,  and  19. 

If  we  pass  a  cut  close  to  a  panelpoint  and  apply  the  inner  forces 
to  establish  the  equilibrium,  then  these  forces  can  be  represented 
by  their  resultant  P,  which  in  turn  can  be  resolved  into  an  axial 
force  S,  a  transverse  force  Q,  and  a  moment  M.  We  refer  the 
reader  here,  to  what  has  been  said  on  this  subject  in  Chapter  II. 
The  two  moments  for  each  bar  either  act  in  the  same  or  in  the 
opposite  sense.  If  in  the  same  sense,  the  deformed  bar  is  either 
on  one  side  of  its  chord  or  partly  on  one  and  partly  on  the  other 
side  of  the  chord,  and  in  both  cases  that  place  where  the  line  of 
the  resultant  P  intersects  with  the  elastic  line  is  a  point  of  inflection. 
Should  the  two  moments  act  in  opposite  directions,  then  the 
deformed  bar  is  on  one  side  only  of  its  chord,  and  it  has  no  point 
of  inflection.  The  resultant  P  intersects  with  the  prolongation 
of  the  chord  of  the  bar  ab. 


22  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


Fig.  14. 


Fig.  15- 


S-. 


Fig.  16. 


COMPRESSION  MEMBERS. 


MANDERLA'S    METHOD 


Fig.  17. 


Fig.  18. 


Fig.  19. 


TENSION  MEMBERS. 


24  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

For  tension  members,  the  convex  side  of  the  elastic  line  is  turned 
towards  the  line  of  the  resultant  P,  and  consequently  no  moment 
between  the  end  points  a  and  b  of  the  bar  can  be  the  maximum 
moment. 

For  compression  members,  the  concave  side  of  the  elastic  line 


V 

Fig.  20. 

is  turned  towards  the  line  of  the  resultant  P,  and  a  maximum 
moment  between  the  ends  a  and  b  of  the  bar  is  possible. 

We  will  now  turn  our  attention  to  the  deflection  angles. 

Figure  20  represents  a  panelpoint  K  where  four  bars  intersect. 
The  straight  bars  are  shown  in  full  lines,  and  any  angle  included 
between  two  adjacent  bars  KS,  which  had  the  value  <j>  before 
deformation,  becomes  <£  +  A<£  after  deformation.  The  assumption 
is  that  during  the  action  of  the  actual  exterior  loads  upon  the  truss 
each  bar  is  free  to  turn  around  a  frictionless  pin.  But  in  reality 
the  bar  ends  are  fixed  by  riveting,  consequently  they  must  be 
deformed  while  the  truss  deflects,  and  the  angles  formed  by  two 
adjacent  end  tangents  KT  of  the  deformed  bars,  shown  in  dashed 
lines,  must  remain  constant  during  deflection,  or,  in  other  words, 
these  angles  are  the  angles  (/>,  which  existed  before  the  exterior 
loads  began  to  act. 


MANDERLA'S    METHOD  25 

In  Fig.  20,  the  angles  of  deflection  r  are  produced  by  clockwise 
revolution  and  are  to  be  taken  positive. 
From  the  figure  we  have 

e  +  p  -=  &  +  ±p  +  T, 

£  +  <£"  +  <£'  =  </>"  +  A</>"  +  <£'  +  A<£'  +  r", 
£  +  0"  +  </>"+  </>'  -  <£'"  +  A<//"  +  c/>"-f  A</>"  +  c//  +  Ac/,'  +  T///> 
or  r'    =  £  -      A0', 

T"        :=     6-      {A^     +     A^i, 

i7"  ==  e  -  {A^  +  A</>"  +  A^}, 
and  generally 

t  -  ?.-  XA^-  (5) 

This  shows  that  if  we  knew  only  one  deflection  angle  £,  then 
from  it  and  the  A<£  all  others  around  a  panelpoint  could  be 
readily  calculated. 

i.    Compression  Members. 

Our  next  step  consists  in  seeking  a  relation  between  the  moments 
M  and  the  angles  of  deflection  T,  and  for  this  purpose  we  employ 
the  equation  of  the  elastic  line,  which  we  write,  bearing  in  mind 
the  directions  of  the  axes  of  the  coordinates  as  shown  in  Figs.  14 
to  19, 

d2y      _  Mx 

doc2         El 

where   E  =  modulus   of   elasticity,  /  =  moment   of   inertia  and 
Mx  —    a  moment  at  any  point  of  the  axis  of  the  bar,  that  is, 

Mx  =  +  Sy  -  Qjc  +  M,. 

5 

Substituting  and  putting  the  three  constants  —  =  T2, 

El 


wehave  ^  =  -  Ty  + 

and  by  integration 

(6) 


26  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

The  constant  B  is  determined  from  the  condition  that  y  becomes 
o   for   x  =  o,    consequently 


The  constant  A  is  found  by  putting  y  =  o  for  x  =  I,  that  is, 

M         p  i 

A  =-'-  tan  -  —  /  tan  to—  -  • 

S  2  sin  p 


cu 


.     If 


/  S 
In  this  last  equation  p  =  IT  •-=  I    y  —  ,  and  tan 

we  put  in  equation  (6)  the  values  for  the  constants  A  and  B,  we 
get  by  differentiation, 

dy     Mtr(,      p  .  )  (  p  ) 

<  tan  —  cos  xl  —sin  xl  }  +tan  oj  <  i  -        —cos  xl  >    (7) 
dx       S    (        2  )  (         sin  p  y 

The  equation  (7)  gives  the  angles  of  deflection 

dy  ,  ,  dy  , 

--  ==  rl  for  x  =  o  and  --    =  ±  r2   for  rv  =  /. 


These  angles  are : 
M 


T  tan    •    -  tan  6; 

2  sn  /o 


-  i 


±  T,  =  - 


tan        -  tan 

2 


P  ) 

-      -  cos  p  -  i 
sm  p  ) 


(8) 


If  we  now  put  for  brevity's  sake, 


2    — 


+  cot  —  -  2  ?Mr  and 


p  p 

—  —  cot  —  =  2  n, 


2  —  p  cot  — 
2 


or 


2  —  /?  COt   L 

2 


MANDERLA'S    METHOD  27 

we  obtain  by  addition  of  the  equations  (8), 


f         (.  f 
tan  co  -      — {-,  d    T2},  (9) 


and  by  subtraction, 


S 
l  ==  -  X-~i  ±  neT2}y  (10) 


and  by  exchange  of  the  angles  r, 

M2  =|{±  mcT2  +  n,-,}.  (ID 

For  practical  calculations  it  is  best  to  express  mc  and  wc  in  series 
as  follows: 

1        P      P3 
cot  p  =  -  - 

P       3      45 

cot£-.2-£-  -f 

2/06       360 


2  -  p  COt 

2 


6    (  60       2520 

6  i           i       a 

-  -  p- , 

p-  10       1400 


42  II 

W,;   =  -    p  -  -     p3 

p          15  6300 

nc  =  ~  +  —  P  +  -±2-  p 
P       30          12600 


2.    Tension  Members. 

The  moment  Mx  at  any  point  of  the  axis  of  a  tension  member  is 

Mx  -    -Sy- 
and  consequently 


£/        £7 


28  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


By  integrating  we  find 


M 
-*•-' 


y  =  A  £   xT  +  BexT  —  tan  <D  .  x  +  —~  • 

o 

y  becomes  o  for  x  =  o,  or 

A  =  ^ l-     -  B  and  B  =  -~  —  A. 

iu  O 

As  y  =  o  for  x  =  I,  we  get  the  values  of  the  constants 

MA    i  -  ep    )   .   j.  i 

A.  =  -p        \~  r    i   *  tan  &>  "~  j 

5    <    £p   -    £~p)  (f   ~    £~p 


(12) 


Of  course  we  have  here  again 


T2EI  =  Sj  ~ 1  =  tan  o>  and  /T1  =  p. 

After  substituting  the  values  of  the  constants  A  and  B  in  the 
equation  (12),  we  obtain  by  differentiation  the  angle  of  deflection 

dy_ 

dx  = 

&L  _      ^xT  J    i  --  £p 

^~  5       ?     £p    -    £-p' 


+ 


+  tan  & 


rrxp  -  i 


(13) 


Further, 


-T  =  TI  for  5f  =  o  and 
rfjc 


r2  for  x  =  I, 


or 


: 

£p  +  £~p 

\ 

Jft 

1 

2 

e 

TI           61  " 

£p    -    £~P 

£p    -    £-p 

2 

2 

M 

I 

2 

2 

±  T,  -     -  —  2 

£p    -    £~p 

£p    -    £~p        ^ 

2 

2 

. 

'  (14) 


MANDERLA'S    METHOD  29 

It  is  suitable  for  our  purpose  to  replace  the  exponential  func- 
tions by  hyperbolic  functions,  which  latter  we  express  in  series. 
The  hyperbolic  sine  and  the  hyperbolic  cosine  are  written: 


.       .  -  , 

sm  hp  =   -         —  and  cos  hp  = 
2 


We  have  further, 

7  p        i  +  cos  hp  ,  p       i  —  cos  hp 

cot  h  -  =  -  —J—  and  tan  h  -  =  -  ;  —  ;  —  —  , 

2  sm  hp  2          sm  hp 


and  consequently  the  equations  (14)  are  transformed  into 

(15) 


— J   T  tan  h  ?-  +  tan  co  \  — 9- —  -  i 
5  2  (  sin  hp 


±  r2  =  -          T  tan  A  £  +  tan  cu     -r~~  X  cos  hp-  i 
S  2  sin  /zo 


If  we  write 


+  cot  h  —  =  2  mt. 

j     P  2 

p  COt  fl   —    —2 


2 


p 

-  cot  h  —  =  2  nt, 


i    P  2 

p  cot  h  -     -2 

2 

we  find  by  addition, 

i  m  +  n 


p  cot  h  -   -  2 

2 


But 


•  7  •  *  x\      ^v^>j       /i/r'  j.  , 

sm  hp  sm  /z/o  sm 

P  n 

=  /O  COt  /Z--  -   2   = 


2  mt  +  nt 


30  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

and  in  adding  the  equations  (15)  and  using  this  last  expression  we 
obtain 

tan^=^^{r1±T2},  (16) 

P 

and  in  subtracting  them, 

£ 
Ml=-  {mt  TI  ±  «,  r2},  (17) 

and  by  exchanging  the  angles  r, 

5 
M2  =  —  {  ±  w,  r2  +  «,  TJ}.  (18) 

We  express  now  cot  hp  in  a  series  that  is 

cot/2/0=I  +  £_^+2-£L 
^       3      45      945 

and  consequently  we  have  in  a  similar  manner  as  that  for  com- 
pression members, 


T  p  p        10       1400 

p  cot  hL  -  2       r 

2 


15       6300 

p      30      12600* 

3.  Determination  of  the  Deflection  Angles  from  the  Conditions 
of  Equilibrium. 

The  final  step  consists  in  writing  down  the  equations  from  which 
the  unknown  deflection  angles  are  to  be  computed,  and  in  showing 
the  manner  in  which  these  calculations  are  effected. 

With  the  knowledge  of  the  values  of  the  deflection  angles  the 
problem  is  really  solved,  as  the  remainder  of  the  computations 


MANDERLA'S    METHOD 


refers  to  simple  operations  with  equations  already  known.     With 
this  end  in  viewr,  we  consider  the  equations: 


As 


Ml=—  \mcrl  ±  ncr2 

S 
M2  =  --  I±mcr2  +  nc 


M2  =  |j±™(r2  + 

5  _  s  VEI 

T"     VS 


for  compression  members. 


for  tension  members. 


j  we  write 


3° 


12600 


^4  2  II 

mt-=K=  }  *-  +  --p  -         ^ 

r          (p     15       6300 


r 

nt-  =  L=  ]  - 

T  (p      30 


12600 


r 

tor  com- 

\ 

pression 

'  V  F  T  9 

r 

members. 

|V£/5 

for  ten- 

sion 

I  VEIS 

members. 

The  general  equation  for  the  moment  M1}  which  is  the  moment 
at  the  end  a  of  any  of  the  bars  in  Figs.  14-19,  is 


S 
=--  —<  mr,  ±  nr2  I    ••=  Kr,  ±  Lr2) 


(19) 


in  which  the  letter  c  or  /,  pointing  to  compression  or  tension,  has 
been  omitted,  in  consequence  of  which  m,  n  may  refer  to  either  a 
compression  or  tension  member;  it  depends  on  what  bar  is  under 
consideration,  and  with  this  understanding  the  equation  will  be 
used. 

We  now  consider  that  in  Fig.  21,  the  bar  ends,  formerly  called  o, 
and  4  in  this  particular  case,  all  meet  in  the  panelpoint  c.     We 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


give  further  to  each  moment  the  positive  sign,  when  it  turns  in  the 
same  direction  as  that  of  a  hand  of  a  clock,  as  is  shown  by  the 
arrows,  which  indicates  that  the  angles  of  deflection  are  produced 
by  clockwise  revolution.  If  the  result  of  the  computation  shows  a 


positive  sign  for  a  moment,  then  it  turns  as  shown  in  the  figure; 
and  if  a  negative  sign,  it  turns  in  the  opposite  direction. 
The  equilibrium  of  panelpoint  o  requires  that 

4 

%M  =  o-  (20) 

i 

If  we  substitute  in  equation  (20)  the  4  moments  of  the  form  of 
the  equation  (19)  by  using  the  notation  as  given  in  Fig.  21,  where 
the  angles  of  deflection  r  appear  either  in  the  form 

£  +  a  or  £  +  /?, 
then  we  have 


01 


# 

#02^0 
#03U0 
#04^0 


ft! 

ft} 


=  o. 


(21) 


Calling 


X 

X 


X 

X 


#04   X    «4    = 

X  ft  = 


•"•"Ol    X     Si     ~T    -^02     X     S2    ~T~    -^03    X     S3       I      -^04    X     £4 


[     UNIVLKoiTY 

OF 


MANDERLA'S    METHOD  33 

the  four  equations  (21)  are  written, 


=  o, 
or 


. 

Each  panelpoint  gives  an  equation  for  a  deflection  angle  £,  and 
as  we  have  just  as  many  unknown  £  as  panelpoints,  the  deflection 
angles  £  can  be  determined.  But  the  computation  of  the  angle  £0 
from  the  equation  (22)  has  a  difficulty,  which  forbids  its  direct 
solution,  and  this  difficulty  consists  in  the  fact  that  the  third  mem- 
ber of  the  numerator  contains  the  four  unknown  quantities  £t,  £2, 
£3,  and  c  4. 

In  order  to  remove  the  difficulty,  we  resort  to  trial  computa- 
tions, and  put  tentatively  ^  =  £2  =  £3  =  £4  =  o,  and  compute 
the  angle  £0  under  this  supposition,  which  is  in  so  far  justified,  as 
the  SZ,£  is  very  small  compared  with  the  sum  of  the  others. 

In  a  similar  way  we  proceed  with  the  determination  of  the 
deflection  angles  £  of  the  other  panelpoints,  and  consequently 
also  with  that  of  £l}  £2,  £3,  and  £4,  and  substitute  these  latter  values 
in  equation  (22),  which  now  yields  a  more  precise  value  of  £0,  and 
in  this  way  we  continue  the  operations  until  satisfactory  results 
are  obtained.  After  these  values  for  c  are  known  they  are  sub- 
stituted in  equation  (5),  and  then  the  deflection  angles  r  are  com- 
puted, and  finally  the  values  of  r  thus  found  are  substituted  in 
equation  (19),  whereupon  the  moments  M  are  obtained.  If  d  is 
the  distance  between  the  neutral  axis  and  the  extreme  fiber,  we 
find  the  secondary  stress 

Md 
T 

The  position  of  the  lines  of  direct  stresses  in  the  bars  is  an  easy 
matter  to  determine  after  the  values  M  are  known.  Considering 
that  the  angle  at,  which  is  included  between  the  resulting  force  P 


34  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

and  the  chord  of  the  deformed  bar,  is  always  very  small,  we  commit 
no  material  error  in  putting  P  =  S,  so  that  we  have, 

Ml          A    ,       M2 
!>--,    and /,=;-'• 

If  we  make  ample  provision  against  buckling  in  the  design  of 
the  truss  members,  that  is,  if  we  design  them  with  large  moments 

of   inertia,    then   p  =  IT  =  I  V  — ;  becomes  very  small,  and  the 

*  hi 

equation  (19),  assuming  a  double  curvature  so  that  both  deflec- 
tion angles  rl  and  -2  are  positive,  is  then  written 


and  for  M 2  we  get 


After   some   simple   transformations   we   find   from   these   two 
equations  the  values  of  the  deflection  angles 


l{2  M ,  -  J/J 

6  El 


6  El 


(23) 


These  equations  (23)  are  identical  with  the  equations  (2)  in 
Chapter  II,  which  latter  we  developed  by  applying  the  principle 
of  the  derivative  of  work. 

The  derivation  of  the  equations  (2)  is  based  on  the  assumption 
that  the  deformation  of  a  bar  compared  with  its  dimensions  is  very 
small  and  consequently  a  negligible  quantity,  but  the  foregoing 
analysis  demonstrates  that  this  assumption  is  only  justified  for 
large  moments  of  inertia.  Therefore,  the  equations  (23)  or  (2) 
should  not  be  employed  for  slim  and  flexible  members. 


CHAPTER    IV. 

MULLER-BRESLAU'S    METHOD. 

i.     Derivation  of  the  Fundamental  Equations. 

THIS  method  neglects  the  influence  of  the  deformation  of  the 
bar  on  the  secondary  stresses,  which  is  justified,  as  we  have  seen 
in  Manderla's  method,  for  sufficiently  large  moments  of  inertia. 

The  method  proceeds  from  the  assumption  that  the  exterior 
loads,  for  which  the  secondary  stresses  are  to  be  computed,  are 
exclusively  applied  at  the  panelpoints  and  produce  a  deformation 
of  the  axis  of  each  bar  showing  no  point  of  inflection,  as  indicated 
in  Fig.  22. 

This  assumed  deformation,  made  for  the  sake  of  determining 
the  character  of  the  signs  of  the  bending  moments,  does  not  exist 


Fig.  22. 


Fig.  23. 


in  reality,  because  the  sum  of  the  alterations  of  the  angles  a,  /?, 
and  7  must  vanish,  that  is  to  say,  we  must  have 
Aa-  +  A/?  +  AY  =  o. 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


For  instance,  should  the  moment  M^  be  found  negative,  then 
the  deformation  of  the  triangle  is  that  of  Fig.  23. 

The  alterations  of  the  angles  are  to  be  calculated  according  to 
equations  (i)  in  Chapter  II,  under  the  assumption  of  frictionless 
pins,  but,  as  the  bar  ends  are  supposed  to  be  rigidly  fixed  by  rivet- 
ing, the  angle  alterations  are  conceived  as  reduced  by  the  bending 
moments  to  their  original  values  a,  /?,  and  7,  as  previously  explained 
in  Chapter  II. 

The  equations  (3)  are  those  for  the  deflection  angles  of  a  bar 
with  single  curvature,  and  they  are  written  with  regard  to  the 
designations  as  given  in  Fig.  22, 


2M. 


El, 

M-\k' 


(24) 


in  which  llt  12,  713  72,  are  the  lengths  and  moments  of  inertia  of  the 
bars  AC  and  AB.     M  denotes  a  bending  moment  and  E  is  the 
modulus  of  elasticity. 
We  have  further, 


(25) 


If  we  substitute  the  values  r2,  r3  .  .  .  .  from  equations  (24)  in 
equations   (25)  we  find 


{M,  +  2  M2} 


{2 


M}        = 


{M3  +  2  M,}  -      +  {2  M5  +  M,}  -      =6  £A/?, 


{M,  +  2  MJ      -  +   {2  M!  +  M2}     -  =6 


(26) 


If  we  assume  for  the  present  that  three  moments  are  known,  we 
would  then  be  in  a  position  to  calculate  the  other  three  bending 
moments  from  equations  (26).  Later  illustrations  will  show  the 


MULLER-BRESLAU'S    METHOD  37 

manner  in  which  the  supposedly  known  moments  are  found. 
Now  for  the  sake  of  convenience  we  do  not  introduce  the 
unknown  moments  M  in  the  computations,  but  the  expressions 


M  -  =  U,  and  put 


for  the  bar     AC  =  I,  : 


"     "      "       AB  =  12  :  M3       ==  Ua  and  M4f-  =  U4. 

12  12 

"    "      "       BC  -  /,  :  M543-  =  U.  and  Me^  =  Ur 

^3  ^3 

With  these  expressions  inserted  in  equations  (26)  we  obtain 

U,  +  2  {U2  +  Z7J+  Z74  =  6EA«.  (27) 

J73  +  2  {U4  +  t75 }  +  U9  =  6  £A/?.  (28) 

^5  +  2  {t/6+  Z7J+  t/2  =  6£A7.  (29) 

We  have  further  the  relation 

U,  +  U2  +  U3  +  U4  +  U5  +  U,  =  o,  (30) 

because  the  sum  of  the  alterations  of  the  angles  a,  /?,  and  7  equals 
zero,  that  is,  Aa  -f  A^  +  A7  =  o.  We  now  express  U4,  U5  and 
t/6  as  functions  of  the  assumed  known  quantities  U^  U2,  and  Z73, 
and  obtain  from  equation  (27), 

U4  =  6  E±a  -  U,  -  2  {Z7,  +  U3},  (31) 

and  from  equations  (28)  and  (30), 

U&  =  6  E±t8  +  Ul  +  U2  -  U4,  (32) 

and  from  equations  (29)  and  (30), 

U,  =  6  £A7  -  Z7,  +  t/3  +  ^  (33) 

As  soon  as  the  quantities  U  are  known,  the  bending  moments  M 

are  also  known,  and  with  them,  of  course,  the  secondary  stresses; 

but  in  order  to  find  these  stresses  in  a  truss,  it  is  necessary  to  apply 

the   foregoing   equations   successively   to   the   different   triangles, 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


which  compose  the  truss,  whereby  the  triangles  are  supposed  to  be 
alternately  deformed  according  to  Fig.  22  or  Fig.  24. 

In  order  to  calculate  the  alterations  of  the  angles  in  those  tri- 
angles deformed  as  in  Fig.  24,  we  must  bear  in  mind  to  reverse  the 
signs  of  A  a,  A/?,  and  A  7,  and  to  write 


sl  -  s3\  cot  7 


Is, 


cot  a  + 


-  **} 


=  o. 


cot  p 
cot  7 


BAA 


(34) 


2.     Determination  of  the  Quantities  U  for  a  Truss. 

In  calculating  the  quantities  U  for  a  truss,  Fig.  25,  we  assume 
for  the  present  that  the  values  U^  and  U2  are  known,  and  we  then 
express  every  other  U  as  a  function  of  U1  and  U2.  The  equilibrium 
requires  that  for  every  panelpoint  Slf  =  o,  consequently  we  have 
at  panelpoint  A : 

M2  —  M3  =  o,  or  U3-~    =  Uz-r-- 

12  /! 

After  Z73  is  known,  its  value  is  substituted  in  equation  (31)  and 
£7  4  calculated,  whereupon  U5  is  found  from  equation  (32)  after  the 
substitution  of  Z74,  and  in  a  similar  way  UQ  is  found  from  equa- 
tion (33). 


MULLER-BRESLAU'S    METHOD 


39 


At  panelpoint  B  the  value  U7  is  determined  from  the  condition 
that 

MQ  —  MI  —  M7  =  o, 

L 


or 


In  going  over  to  the  triangle  called  II  in  Fig.  25,  we  simply 
repeat  the  calculation  made  for  triangle  I,  that  is  to  say,  we  write 
down  the  equations  for  Us)  U9,  and  Ulo  in  conformity  with  equa- 
tions (31),  (32),  and  (33),  care  being  taken  that  the  alterations  Aa, 


U13 


A/?  and  AY  are  calculated  for  triangle  /  after  equations  (34)  and 
for  triangle  II  after  equations  (i).  Knowing  U4,  U5  and  t/10,  we 
find  Un  at  panelpoint  C  from 

-  M4  -  M,  +  Mlo  -  Mu  =  o, 

and  finally  U12,  U13  and  U14  are  found  from  equations  which 
again  correspond  to  the  fundamental  equations  (31),  (32),  and  (33). 
We  have  now  expressed  each  U  as  a  function  of  Ul  and  £72, 
which  were  supposed  to  be  known,  and  the  next  step  consists  in 
determining  Ul  and  U2.  To  this  end  we  apply  the  condition 
2M  =  o  to  the  panelpoints  D  and  E,  and  write 


LL  _ 


=  o. 


40  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

The  solution  of  these  equations  gives  £/x  and  U2)  and  herewith 
the  problem  is  solved. 

3.     Method  of  Influence  Lines. 

The  secondary  stresses  are  usually  computed  for  one  single 
position  of  the  loads.  A  more  complete  investigation  would  con- 
sist in  the  computation  of  these  stresses  under  the  assumption  of 
full  load  and  one  or  more  partial  loadings,  but  the  most  complete 
investigation  applies  the  method  of  influence  lines. 

By  Manderla's  method  we  have  seen  that  the  values  K  and  L 
are  higher  functions  of  the  primary  stresses  5,  and  as  the  secondary 
stresses  are  dependent  on  K  and  L,  they  are  also  higher  functions 
of  S  or  of  the  exterior  loads,  in  consequence  of  which  the  employ- 
ment of  the  method  of  influence  lines  is  not  to  be  thought  of. 

It  was  also  shown  that  if  the  influence  of  the  deformations  of 
the  bars  is  left  out  of  consideration,  then  the  resulting  equations 
are  the  more  exact  the  larger  the  moments  of  inertia  or  the  stiffer 
the  truss  members  are,  and  in  this  case  the  secondary  stresses  are 


found  to  be  linear  functions  of  the  exterior  loads,  and  the  method 
of  influence  lines  is  applicable. 

In  using  this  method,  an  influence  line  may  be  drawn  for  each 
bar  end  under  the  assumption  of  a  traveling  load  equal  to  unity 
which  is  successively  applied  to  the  panelpoints  of  the  truss. 


MULLER-BRESLAU'S    METHOD  4! 

Muller-Breslau's  method  of  influence  lines  is  in  substance  as 
follows  : 

Figure  26  represents  a  truss  composed  of  the  triangles  I  to 
VII,  the  sides  of  which  show  alternating  deformations,  and  whose 
angle  alterations  must  be  calculated  alternately  in  compliance  with 
equations  (i)  and  (34).  The  values  Ult  U2,  U3,  etc.,  correspond 
to  the  figures  i,  2,  3,  etc.,  written  near  the  bar  ends  in  Fig.  26. 

The  first  step  consists  in  the  calculation  of  the  angle  alterations 
after  a  load  is  applied  to  the  nearest  panelpoint  of  the  right-hand 
support  —  in  our  case  panelpoint  (4)  —  and  of  such  magnitude 
that  it  produces  at  the  left-hand  support  a  reaction  =  i.  For  this 
reaction  :  =  i ,  each  U  value  from  U3  to  U2T  inclusive  is  expressed 
as  a  function  of  U1  and  U2  in  a  manner  as  has  been  previously 
explained.  Thereupon  the  load  is  shifted  from  panelpoint  (4)  to 
panelpoint  (i),  and  it  is  given  such  a  magnitude  that  it  produces  at 
the  right-hand  support  B  a  reaction  =  i. 

The  calculations  must  now  be  repeated.  First,  all  angle  altera^ 
tions  are  calculated  for  a  reaction  =  i  at  B,  and  each  U  from  U^ 
to  U4  is  expressed  as  a  function  of  Z730  and  U2r 

If  the  load  is  applied  at  (4)  the  general  expression  for  Um  is 

Um    «    Vm    +   CmU,    +   DmU2J  (35) 

and  if  the  load  is  applied  at  (i)  we  have 

Um  =  Wm  +  FmU30  +  GmUm.  (36) 

Vm,  Wm,  Cm,  Dm,  Fm,  and  Gm  are  known  quantities,  and  Vm  and 
Wm  alone  are  functions  of  the  alterations  of  the  angles  which  are 
dependent  on  the  assumed  exterior  load. 

If  we  now  apply  to  panelpoint  (2)  a  vertical  load  equal  to  unity, 
then  this  load  produces  a  reaction  at  the  left-hand  support  equal  to 

Ra  =  i  X  j,  and  the  stresses  due  to  this  reaction  in  the  sides  of  the 

JL/ 

triangles  marked  I  and  II  are  —    times  greater  than  those  pro- 

JL/ 


42  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

duced  by  the  reaction  Ra  =  i,  in  consequence  of  which  the  equa- 
tion (35)  is  transformed  into 


U, 


DmU. 


(37) 


The  value  £/n  is  also  computed  after  equation  (37)  for  the 
simple  reason  that  the  condition  £M  =  o  must  be  fulfilled,  that 
is  to  say,  we  must  have 

-  Mn  +  M10  -  M,  +  M4  =  o, 
or,  by  multiplying  with  —  i  and  writing  U  —  instead  of  M,  we  get 

i 


utl  f  -  u, 

^8 


/ 


The  same  reasoning  holds  also  true  for  the  right-hand  reaction 
Rb,  the  stresses  and  angle  alterations  in  the  triangles  IV,  V,  VI, 
and  VII,  so  that  we  are  in  a  position  to  write  down  the  following 
set  of  equations  : 


U    - 


+C8C7,    +DSU 


S2 


U,   =      V,    +  C,  L\    +  D,  U, 


U, 


U     --W 

U  13    ~    j    VV  13 

&  T1 


(38) 


MULLER-BRESLAU'S    METHOD  43 

With   respect    to  equations    (32)  and  (30)    and    the   condition 
M  =  o,  applied  to  panelpoint  (2),  we  now  write  down: 


U13  +  U14, 


U9    +  U1 


(39) 


After  the  values  U8  to  Z715,  as  expressed  in  equations  (38),  are 
inserted  in  equations  (39),  they  can  be  solved  for  Uly  U2,  U30  and 
t/29.  In  applying  the  load  =  i  in  succession  to  the  different 
panelpoints  of  the  truss,  we  obtain  by  the  same  course  of  treatment 
the  influence  lines  for  U1}  U2,  t/30  and  U.M,  and  with  these  latter 
values  the  influence  lines  for  any  U. 

We  caution  the  reader  to  observe  the  fact  that  Aa'm  and  A/?,,,  in 
equations  (39)  require  a  separate  calculation  for  the  reason  that, 

if  the  load  =  i  is  supposed  at  (2),  then  the  reaction  Ra  =  i  X  - 

produces  -  times  greater  stresses  and  angle  alterations  than  Ra  =  i 

in  the  triangles  I  and  II,  but  not  in  triangle  III. 

For  each  U  we  have  two  equations,  the  use  of  which  depends 
on  the  position  of  the  load.  If,  for  instance,  the  load  is  applied 
either  at  panelpoint  (3)  or  (4),  then 

Ua  =  2V13  +  CnUt  +  D13U2; 
but  if  the  load  is  at  (i)  or  (2),  the  equation 

77  =-W  +  F  U  +  G  U 
must  be  used. 


CHAPTER    V. 
RITTER'S   METHOD. 

OUR  first  step  will  consist  in  assigning  the  proper  notation  for 
the  different  quantities  with  which  we  have  to  deal. 

In  Fig.  27,  representing  a  part  of  a  truss,  we  will  for  the  present 
exclusively  consider  panelpoint  (5)  where  four  bars  intersect, 
forming  three  angles.  Each  of  these  angles  included  between  any 


Fig.  27. 

two  adjacent  straight  bars  will  be  denoted  by  an  index  corre- 
sponding to  the  two  figures  of  the  opposite  bar.  So,  for  instance, 
the  £  3-5-4  shall  have  the  index  3-4,  the  ^  4-5-6  the  index  4-6, 
etc. 

The  two  bending  moments  of  the  bar  3-5  will  be  designated 
M3  at  panelpoint  (5)  and  M3'  at  panelpoint  (3).  For  the  bar  4-5 
the  moments  shall  be  Af4  at  panelpoint  (5)  and  Mf  at  panelpoint 
(4),  etc.  But  in  case  we  consider  panelpoint  (4)  the  designations 
of  the  two  moments  of  the  bar  4-5  are  M5  at  panelpoint  (4)  and 
MK'  at  panelpoint  (5).  Consequently  we  can  write: 

M4  for  panelpoint  (5)  =  M&f  for  panelpoint  (4), 
MI   for  panelpoint  (5)  =  M5  for  panelpoint  (4), 

44 


RITTER'S    METHOD  45 

which  means  that  each  moment  in  a  truss  is  characterized  in  two 
different  ways.  A  moment  will  be  taken  as  positive  when  it 
deflects  a  bar  in  the  sense  of  the  hand  of  a  clock. 

If  each  bar  in  our  truss  could  turn  around  a  frictionless  pin, 
then  each  angle  a  included  between  any  two  adjacent  bars  would 
be  changed  under  the  influence  of  the  actual  loading  of  the  truss; 
these  angles  would  be  either  increased  or  decreased  an  amount  A  a 
corresponding  to  the  alterations  in  the  lengths  of  the  bars.  But 
the  bar  ends  of  our  truss  are  riveted,  consequently  the  angle  included 
between  any  two  adjacent  end  tangents  drawn  to  the  curves  of  the 
deformed  bars,  which  we  assume  to  be  S-shaped,  must  remain 
unchanged  during  deformation,  and  this  unchangeable  angle  is  a. 
We  refer  the  reader  here  to  Fig.  i  in  Chapter  II,  and  to  what  has 
been  further  said  on  this  subject. 

Figure  27  shows  that 


or 


Substituting  the  values  of  r3  and  r4  as  given  in  equations  (2)  of 
Chapter  II,  we  have: 

/3J2M3-M/J       /4{2M4-M/j 
-4=          .    6/3  ~67T 


Similar  equations  can  be  written  for  EAa4_0  and 

We  put  now  for  the  sake  of  convenience    -  -  =  U,  U  repre- 

6  1 

senting  a  force  per  unit  of  area  and  measured  with  the  same  unit 
as  s  and  EAa.  The  equilibrium  at  a  panelpoint  requires  that 
the  algebraic  sum  of  the  moments  vanishes,  or  that 

M3  +  M4  +  M8  +  M7  =  o, 

and  the  four  equations  in  regard  to  panelpoint  (5)  are  now  as 
follows  : 


46  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


.<  =  {2  17.-  17,'}  --  \2Ut-  Ut'}, 
..-  J2t/4-  [//}  --  J2tf.  -  17.'}, 
.,  =  J2^c-  f/0'j  --  \2U.  ~  U/}. 


(40) 


For  every  panelpoint  there  are  as  many  equations  as  bars, 
which  intersect  at  the  panelpoint,  or  two  equations  for  each  bar, 
so  that  the  total  number  of  equations  equals  the  number  of  unknown 
moments. 

The  work  of  solving  a  larger  set  of  equations  algebraically  taxes 
the  patience  of  a  quick  and  sure  computer,  experienced  in  all  the 
short  cuts  that  can  be  advantageously  used,  and  any  means  which 
are  calculated  to  save  time  are  very  welcome  indeed. 

For  this  reason  we  will  now  consider  a  graphical  solution  of 
the  problem,  the  object  being  to  find  the  values  U. 

The  EAa  are  computed  by  means  of  equations  (i)  in  Chapter  II, 
and  are  due  to  the  actual  loading  of  the  truss,  for  which  loading  we 
intend  to  determine  the  secondary  stresses.  If  we  assume  for 
the  sake  of  an  illustration  that  besides  the  EAa  also  the  values  of 
V  were  known,  we  could  then  easily  and  quickly  find  the  values 
U  by  the  simple  means  of  a  force  and  string  polygon  in  the  follow- 
ing manner: 

We  consider  the  values  —  as  forces,  lay  out  a  vertical  load  line, 

i 

select  an  arbitrary  pole  O5  with  the  pole  distance  H5,  and  draw 
the  rays  as  in  Fig.  28.  Hereupon  the  distances  between  the  forces, 
which  are  the  known  EAa,  are  laid  out  horizontally  as  in  Fig.  29, 
and  each  of  the  forces  displaced  to  the  left  an  amount  Uf  and  the 
equilibrium  polygon  constructed  for  the  forces  thus  displaced. 

The  double  values  of  U  are  now  equal  to  the  distances  of  the 
displaced  forces  from  their  resultant,  a  statement  which  follows 
from  the  equations  (40). 

The  distance  of  a  displaced  force  from  the  resultant  is  2  U  for 
a  displacement  U',  consequently  the  distance  of  an  undisplaced 


HITTER'S    METHOD 


47 


force  from  the  resultant  equals  2  U-U',  and  the  difference  between 

the  distances  of  two  adjacent  undisplaced  forces  must  equal  E&a. 

The  meaning  of  this  is  that  the  first  three  of  the  equations  (40) 


Fig.  29. 


have  been  satisfied,  and  since  the  string  polygon  shows  that  the 
algebraic  sum  of  the  component  moments  in  reference  to  any 
point  in  the  direction  of  the  resultant  vanishes,  the  last  equation 
is  also  satisfied. 

As  a  matter  of  fact,  the  values  of  Uf  are  not  known  at  all,  and 
herein  of  course  lies  an  obstacle  to  our  solution,  but  this  can  be 
removed  by  trials. 

Considering  the  fact  that  any  change  in  the  values  of  Ur  has 
only  half  the  effect  on  the  values  of  £7,  we  will  for  the  first  trial 
assume  that  the  quantities  Uf  are  non-existent,  in  which  case  we 
obtain  roughly  approximate  values  U\  or,  in  other  words,  for  the 
first  trial  we  will  assume  that  the  forces  are  not  at  all  displaced 
and  with  this  understanding  we  draw  a  force  and  equilibrium 
polygon  for  every  panelpoint. 

Figures  30-31  show  the  positions  of  the  resultants  of  the  undis- 
placed forces  for  the  panelpoints  (4)  and  (5),  the  equilibrium 
polygons  being  omitted. 


48 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


If  now  the  distance  between  an  undisplaced  force  and  the  result- 
ant is  designated  by  F,  we  have,  with  reference  to  Figs.  30-31, 

2  U,  =  U.'  +  7,, 
a  U,  -  Ut'  +  Vt. 

In  compliance  with  the  designations  as  adopted  at  the  beginning, 
we  have  also  77  '_ 

u['  =  u" 

6534  3  467 


1,111        I 


2U4 


Panelpoint  4 
Fig.  30- 


Panelpoint  5 

Fig.  31. 


Further, 


TJ,    -         > 

U  5      " 


or 


and  in  a  similar  way  we  find 


The  determination  of  the  quantities  U  would  be  an  easy  matter 
if  the  positions  of  the  resultant  forces  were  known,  in  which  case 
they  could  be  found  at  once ;  but  since  these  positions  are  not  known 
and  must  be  first  found,  we  take  for  instance  U3,  which  equals 


RITTER'S    METHOD  49 

and  transfer  it  as  £//  in  the  equilibrium  polygon  for  panelpoint  (5); 
and  in  a  similar  way  we  transfer  any  other  U  as  U',  which  has 
been  obtained  by  a  first  trial,  into  some  other  and  corresponding 
equilibrium  polygon,  and  continue  these  correcting  operations  until 
the  changes  in  the  values  are  so  small  that  they  can  be  neglected, 
but  we  must  bear  in  mind  that  each  equilibrium  polygon  has  to 
be  repeatedly  drawn. 

After  the  quantities  U  are  known,  we  obtain  the  bending  moments 
from  the  equations: 


and  if  d  is  the  distance  from  the  center  line  of  gravity  to  the  extreme 
fiber,  and  cr  the  secondary  stress  per  square  unit,  then 

Md       6dT1 
~      ~TU' 

Strict  attention  must  be  paid  to  the  character  of  the  signs  in 
order  to  avoid  mistakes.  The  succession  of  the  bars  around  a 
panelpoint  should  be  taken  in  the  sense  of  motion  of  the  hand  of 
a  clock,  and  a  Aa,  which  has  a  positive  sign,  should  be  laid  out 
on  the  right  hand,  and  a  negative  A  a  on  the  left  hand.  Any 
quantity  U  to  be  transferred  as  U'  into  a  corresponding  equilibrium 
polygon  should  be  laid  out  to  the  left,  if  it  is  situated  to  the  left  of 

the  resulting  force  of  the  -  ,  and  it  should  be  laid  out  to  the  right, 

if  on  the  right  side  of  the  resultant. 

The  method  we  have  explained  is  in  so  far  approximate  as  the 
influence  of  the  deformation  on  the  secondary  stresses,  due  to  the 
action  of  a  force  along  the  chord  of  the  deformed  bar,  has  not  been 
taken  into  account;  and  if  this  longitudinal  force  is  also  to  be  con- 
sidered, it  is  necessary  to  develop  an  expression  for  the  deflection 
angle  r  differing  from  that  given  in  equations  (2)  in  Chapter  II. 


OF  TM£ 

UNIVERSITY 


50  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

The  equation  of  the  elastic  line  with  the  designations  as  given  in 
Fig.  32  is  written 


doc2 


El 


Mx  being  the  moment  at  any  arbitrary  point  of  the  axis  of  the  bar 
with  the  coordinates  x,  y. 

If  it  is  now  supposed  that  the  bar  is  acted  upon  not  only  by  the 


^  +x 


two  end  bending  moments  M  and  M',  but  also  by  a  compressive 
force  of  the  magnitude  S,  then 


Mx  =  Sy  + 


M\l  -  x} 


Substituting  this  value  M x  in  the  differential  equation,  wre  get 

+  M'\  x  -  Ml 


— tL    —  _   —*L      i 

dx>          El 


EIL 


The  integration  gives 


M 


y-ycos 


Kx       McosK  +  M'     .     Kx 
sin  - 


SsmK 


+  M'}x-Ml 
SI 


In  this  equation  the  constants  have  been  determined  from  the 
condition  that  y  vanishes  when  x  =  o  and  when  x  =  /,  and  K 

/SI2 
denotes  the  expression  y  —  for  brevity's  sake. 

tLL 


RITTER'S    METHOD  51 

The  first  differential  coefficient  gives  the  angle  of  deflection  r 
for  x  =  o,  that  is, 


dy  =  _  K{McosK  +  M'\       M  +  M' 

dx  SI  sin  K  SI 

Developing  cos  K  and  sin  K  in  series  gives 


=  l_\2RM  -  R'M' 


6  El 

and  in  this  last  equation 

2K4   .     K" 


E> 


J5        3*5        1575- 

^  =  I+7A3  +  3^  +  -^c 

oo         2520       100800 

If  M  and  M'  are  exchanged  r'  will  be  found.     Should  the  bar 
under  consideration  be  a  tension  member,  then  the  sign  of  K2  in 

SI2 
the  equation  K2    =  —  must  be  reversed  and  R  and  R'  computed 

J^sJ. 

accordingly,  that  is  to  say,  every  second  member  is  negative. 

For  an  infinitely  great  moment  of  inertia  K2  is  reduced  to  zero 
and  R  =  R'  ••  =  i,  which  leads  us  back  to  the  equations  (2)  in 
Chapter  II. 

From  this  consideration  we  conclude  that  the  equations  (2) 
are  the  more  exact  the  larger  the  moments  of  inertia  are,  or,  in 
other  words,  they  should  only  be  used  for  stiff  truss  members 
which  have  ample  provision  against  buckling,  an  assertion  pre- 
viously made  and  which  has  now  been  demonstrated  to  be  true. 

Should  the  more  exact  method  be  used  in  connection  with 
graphic  statics,  it  is  necessary  in  constructing  the  force  polygon  to 

lay  off  the  forces  —  and    displace    them    in    the    equilibrium 

polygon  an  amount  R'U1 ',  whereupon  the  values  2  RU  are  found 
instead  of  2  U. 


CHAPTER    VI. 

MOHR'S    METHOD. 

i.   Determination  of  the  Unknown  Quantities. 

IN  this  method  of  calculation  the  effect  of  deformation  on  the 
leverarms  y,  as  also  that  of  the  transverse  forces  Q,  are  not  con- 
sidered (see  Fig.  2  in  Chapter  II),  for  reasons  previously  given, 
and  only  the  bending  moment  at  each  end  of  the  bar  is  determined. 

If  for  a  truss,  which*  is  composed  of  triangles,  p  denotes  the 
number  of  panelpoints  and  n  the  number  of  bars,  then 

n  =  2  p  -  3; 

and  as  2  unknown  moments  correspond  to  each  bar,  we  have  as 
the  total  number  of  unknown  moments, 

2  n  =  4  p  —  6. 

But  instead  of  introducing  the  unknown  bending  moments  in 
the  calculation,  Mohr  introduces  two  sets  of  angles  on  which  the 
bending  moments  are  dependent,  and  in  so  doing  he  reduces  the 
total  number  of  unknowns  to  3  p  —  3. 

Figure  33  shows  the  unknown  angles  (f>0,  <j>lt  and  001  for  the  bar 
01.  The  lines  o  a  and  i  b  are  parallel  to  each  other,  and  indicate 
the  original  direction  of  the  bar  01,  which  is  that  before  the  ex- 
terior forces  began  to  act.  o  T0  and  i  Tl  indicate  the  end  tangents 
drawn  to  the  elastic  line  of  the  curved  bar  after  deformation  has 
set  in.  The  angle  <p0  included  between  the  original  direction  o  a  of 
the  bar  and  its  end  tangent  o  TQ  is  constant  for  each  bar  end 
around  the  panelpoint  o  during  deformation,  or,  in  other  words, 
(f>0  is  the  angle  around  which  the  end  of  each  bar  revolves  during 
deformation.  This  fact  follows  from  our  assumption  that  all  bar 
ends  are  rigidly  riveted  so  that  an  angle  between  any  two  adjacent 

52 


MOHR'S    METHOD  53 

bar  ends  must  remain  constant  while  a  bar  becomes .  deformed. 
The  angle  ^01  included  between  the  original  direction  o  a  or  i  b  of 
the  bar  and  the  chord  of  the  curved  bar  after  deformation  is  the 
angle  around  which  a  bar  revolves  during  deformation  under  the 
assumption  that  the  truss  is  provided  with  frictionless  pins. 

To  each  panelpoint  of  the  truss  corresponds  an  angle  <£,  and  as 
the  number  of  the  panelpoints  is  />,  we  have  p  as  the  total  number 


of  the  angles  </>;  to  each  bar  corresponds  an  angle  ^,  and  as  the 
number  of  the  bars  is  2  p  —  3,  we  have  2  p  —  3  as  the  total  number 
of  the  angles  ^,  consequently  the  number  of  angles  (f>  and  ^  taken 
together  is  equal  to  p  +  2  p  -  3  =  3  #  —  3,  which  are  now  the 
number  of  unknowns  of  our  problem. 

2.   Determination  of  the  Angles  (p. 

These  angles  $  are  calculated  by  the  equation  in  Chapter  II, 
M([)  =  2  s A/,  where  M  ==  i,  so  that 

i  X  $  =  ZsM 

for  the  assumption  that  the  points  of  supports  are  fixed  in  the 
direction  of  the  reactions.  This  is  the  case  with  respect  to  Fig.  34, 
which  shows  a  Pratt  truss  with  one  fixed  end  and  one  roller  end. 
If  we  wish,  for  instance,  to  determine  the  angle  of  revolution  <p  for 

the  diagonal  in  the  third  panel  from  the  left  end,  a  force  P  =  -  must 

l 


54 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


be  applied  at  each  end  of  the  diagonal  and  at  right  angles  to  its 
axis,  which  gives  a  moment  -    -f  i.     This  moment  produces  a 

downward  reaction  =  -  at  the  left  end  of  the  truss,  and  an  upward 

.L/ 

reaction  of  the  same  magnitude  at  the  right  end.     Hereupon  the 


Fig.  34- 

stresses  s  for  each  bar  produced  by  the  moment  =  +  i  and  the 
reactions  —  are  calculated  and  the  sum  of  the  products  s&l  is 

jL/ 

SI 
formed,  where  A/  =  —  ,  S  being  the  stress  produced  by  the  actual 

I-JA\. 

loading  (for  example,  by  a  railroad  train),  E  being  the  modulus  of 
elasticity  and  A  the  gross  area  of  a  bar. 

3.   The  Bending  Moments  Expressed  as  Functions  of  the 
Angles  <f>  and  d>. 

In  Chapter  II,  it  was  shown  that  the  elastic  line,  whose  equation 

d2y        M 
is  -r^2  =  —  ,  can  be  conceived  as  an  equilibrium  curve,  if  we  put, 

CLOC  JLH. 

for  instance,  El  =  Hy  and  M  =  p\  that  is  to  say,  by  considering 
first  the  term  El  as  a  force  acting  parallel  along  the  chord  of  the 
deformed  bar,  and  considering  secondly  the  bending  moment  M  as 
a  force  per  unit  of  length  of  the  equilibrium  curve  and  at  right 
angles  to  the  chord  of  the  deformed  bar. 


MOHR'S    METHOD 


55 


In  order  to  obtain  at  once  the  proper  signs  for  the  moments  M 
and  the  angles  <f>  —  </»,  it  is  suitable  for  the  purpose  in  view  to  give 
these  moments  and  angles  the  positive  signs,  letting  the  moments 
turn  in  the  sense  like  the  hand  of  a  clock,  and  producing  the  angles 
by  clockwise  revolution  of  a  right  line,  as  is  indicated  in  Fig.  35. 

The  meaning  of  the  angles  <£  —  &  is  the  same  as  previously 
explained  with  reference  to  Fig.  33.  These  angles  are,  of  course, 


M21 


Fig.  35- 


very  small,  but  they  are  shown  in  the  sketch  largely  exaggerated 
for  the  sake  of  a  better  illustration.  The  variable  moments,  con- 
sidered as  vertical  forces  to  whose  action  the  bar  is  subjected,  are 
represented  as  the  difference  between  the  two  triangles  i  —  2  —  F, 
and  F  —  G  —  2.  Their  resultant  forces  Ql  and  Q2  pass  through 
the  centers  of  gravity  of  the  triangles,  distant  one  third  of  the 
length  of  the  bar  from  each  end.  Besides  the  vertical  forces  Ql 
and  Q2  we  have  the  longitudinal  force  El  acting  upon  the  bar. 
The  four  forces  Rlf  R2,  Qt  and  Q2  are  in  equilibrium  and  form 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


the  trapezoid  A  BCD.  The  equilibrium  requires  that  the  alge- 
braic sum  of  the  moments  with  respect  to  any  point  must  be  equal 
to  zero.  In  taking  the  moments  around  the  two  points  A  and  C, 
we  have  then  the  two  equations, 


El  X  AD  -  Q2-  -=  o, 
o 

El  X  BC  -  Q,  -  =  o. 

O 


(42) 


The  lengths  of  the  lines  AD  and  BC,  bearing  in  mind  that  the 
angles  (f>  and  ^  are  very  small,  we  find, 


BC-i 

O 


il 

3 

il 
3 


The  forces  (J^  and  Q2  are 

Ql  =  Afiai  and 


-i(< 

-    I 

3 

M»  - 


Substituting  the  values  of  yl.D,  5C,  Ql  and  Q2  in  the  equations 
(42),  we  have 


.    P 
216  =  °> 

I2 
»»-Q. 


(43) 


Dividing  each  equation  by  —  and  putting——-  =N12,  the  equa- 


tions (43)  are  then  written, 


In  order  to  find  the  moments  Jlf  it  is  necessary  to  determine  next 
the  unknown  angles  (f>  in  equations  (44)  ;  the  values  TV  are  known, 
and  the  angles  ^  have  previously  been  calculated.  As  the  alge- 
braic sum  of  the  moments  with  respect  to  any  point  must  be  equal 


MOHR'S    METHOD  57 

to  zero,  we  take  in  succession  the  panelpoints  of  a  truss  as  the 
centers  of  moments,  and  wrile  down  as  many  equations  as  there 
are  panelpoints  in  the  truss,  which  number  equals  the  number  of 
unknown  angles  <f>. 

With  reference  to  Fig.  34,  these  equations  are  as  follows: 

Panelpoint      i  :  Mj_2  +  M^_z  =  o. 

2  :  M2__,  +  M2_s  +  M2_,  +  M2_4  =  o. 

3  :  M3_,  +  M3_2  +  Ms_s  =  o. 

4  :  Mt_2  +  M4_5  +  M4_7  +  M4_6  =  o. 

5  :  M5_3  +  M,_2  +  M5_4  +  M5_7  =  o. 
6:lfc_4  +M6_r  +  M6_8  =  o.  I 

7  :  M^,  +  M7_4  +  M7_t  +  MT_,  +  M7_,  =  o.  | 

8  :  M8_6  +  M8_.  +  M8_,  +  M8_10  =  o. 


9  :  M9_7  +  M9_8  +  Af0_10+  M9_u  =  o. 

10  :^f10_8  +  M10_9  +M10_U+  M]_     =  o. 

11  :Mn_9  +  MU_10  +  MU_12=  o. 

_+  M12_n=  o. 


The  values  of  the  bending  moments  M  as  expressed  in  equations 
(44)  are  now  substituted  in  equations  (45),  and  the  values  of  the 
angles  <£  are  ascertained  by  solving  the  latter  equations. 

The  last  step  consists  in  substituting  the  values  <£  in  the  equations 
(44),  from  which  now  the  moments  M  can  be  calculated,  &nd  here- 
with our  problem  is  solved. 

Mohr  suggests  a  short  cut  in  regard  to  the  solution  of  equations 
(45),  provided  the  truss  to  be  examined  is  symmetrical,  in  which 
case  the  conditions  of  equilibrium  appear  in  symmetrical  form. 
For  a  symmetrical  truss  the  work  of  solving  the  equations  is  now 
greatly  facilitated  by  determining  first  the  sums  and  the  differences 
of  the  angles  <£,  whose  positions  are  symmetrical  in  respect  to  'the 
center  line  of  the  truss.  In  our  particular  case  we  would  determine 
first 

{<£i  +0i2(  and  {0,  -<£J2},.  -.1 

{<£2  +  c£10(  and  {<£2-<£10}, 
^   +  *       and    ^    -  etc. 


CHAPTER    VII. 

METHOD    OF    LEAST   WORK. 

THERE  is  no  doubt  that  any  subject  gains  in  clearness  if  looked 
at  from  different  points  of  view,  and  for  this  reason  we  will  make 
use  of  the  principle  of  least  work  with  which  the  reader  is  supposed 
to  be  familiar  and  apply  it  to  the  simplest  truss,  composed  of  three 
bars  and  shown  in  Fig.  36. 

TABLE    I. 


Stresses  in  Tons,  if  Pin 

Gross 

Bar. 

Length  in 
Inches. 

Connected. 

Area  in 
Square 
Inches. 

Moment  of 
Inertia,  /, 
Reduced  to 
Inches. 

7 

length' 

Total. 

Per  Square 
Inch. 

AC 

400 

-  176.75 

-6 

29.46 

800 

2 

AB 

565.68 

+   125.00 

+  7 

17.86 

6OO 

1.  06 

BC 

400 

-  176.75 

-  6 

29.46 

800 

2 

The  riveted  truss  is  symmetrical  in  its  form,  loaded  with  a  single 
concentrated  load  of  250  tons  at  its  apex,  and  assumed  to  have 
one  fixed  and  one  movable  end.  The  necessary  data  for  the  cal- 
culations are  given  in  Table  I,  which  are  self-explanatory.  Each 
bar  is  box-shaped,  consisting  of  two  1 5-inch  webs  and  4  angles. 
Before  analyzing  the  stresses,  it  is  necessary  to  consider  the  nature 
of  the  truss  under  consideration. 

We  have  seen  that  the  lines  of  stresses  in  each  bar  of  a  riveted 
truss  are  displaced  under  the  action  of  the  exterior  forces.  Of 
these  stresses  we  know  neither  their  magnitude  nor  their  position 
and  direction;  that  is  to  say,  each  bar  represents  three  unknown' 

58 


METHOD    OF    LEAST    WORK 


59 


quantities.  If  p  designates  the  number  of  panelpoints  and  n  the 
number  of  bars  in  a  truss,  then  n  =  2  p  —  3,  and  consequently  the 
number  of  unknowns  is  equal  to  3  n  =  6  p  —  9.  The  statics  of 
rigid  bodies  gives  us  three  equations  for  each  panelpoint,  so  that 
we  have  a  total  of  3  p  equations;  but  as  this  number  includes  three 
equations  which  refer  to  the  equilibrium  of  the  outer  forces,  and  are 

250  Tons 


consequently  of  no  use  for  our  purpose  to  determine  the  inner 
forces,  there  are  only  3  />  —  3  equations  available  for  the  deter- 
mination of  the  6  p  —  9  unknowns.  The  remainder  of  the  equa- 
tions, that  is,  { 6  p  —  9  j  -  -  {3^  —  3}  ;=  3  p  —  6,  must  therefore 
be  obtained  from  some  other  source  than  statics.  From  the  fore- 
going remarks  we  see  that  a  triangular  riveted  truss  is  threefold 
statically  indeterminate,  a  fact  which  can  also  be  arrived  at  in  a 
way  different  from  the  above. 

The  influence  of  the  sectional  areas  on  the  stresses,  which  always 
exists  in  a  statically  indeterminate  structure,  can  easily  be  detected 
in  our  truss  by  going  to  extremes.  Let  us  suppose  that  the  truss 
is  loaded  at  its  apex  with  a  finite  load,  but  that  the  moments  of 
inertia  of  the  two  compression  members  are  infinite,  then  any 
deformation  of  these  members  is  excluded,  and  as  the  angle  at  the 
apex — the  truss  being  riveted  —  remains  unchanged,  it  follows 
that  the  length  of  the  horizontal  bar  is  not  affected  at  all  by  the 


60  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

exterior  load.  This  is  merely  another  way  of  stating  that  the 
horizontal  bar  is  under  no  stress.  If  we  now  suppose  each  moment 
of  inertia  of  the  two  compression  members  to  be  equal  to  zero,  in 
which  case  each  of  these  two  members  is  represented  by  a  very  thin 
bar,  hinged  at  its  ends,  then  the  horizontal  bar  receives  a  pull  of 
125  tons,  but  no  bending  moment.  As  neither  of  these  conditions 
can  be  fulfilled,  that  is,  as  the  moments  of  inertia  of  the  members 
must  be  between  o  and  x ,  we  draw  the  conclusion  that  the  pull 
in  the  horizontal  bar  is  less  than  125  tons. 

The  circumstances  that  the  truss  is  symmetrical  in  form  and 
also  symmetrically  loaded  reduces  the  three  unknowns  to  two 
unknowns.  Generally  speaking,  we  are  free  to  select  the  unknowns 
of  the  problem,  and  in  our  case  we  take  the  pull  of  the  horizontal 
bar  and  the  bending  moment  at  its  middle  as  the  two  unknown 
quantities  to  be  determined.  The  only  stresses  we  need  to  con- 
sider in  our  case  are  direct  and  bending  stresses;  the  shearing 
stresses,  which  may  be  taken  into  account,  are  in  so  far  of  no  con- 
sequence, as  their  influence  on  the  final  result  is  small  enough  to 
be  neglected,  and  the  effect  of  the  moments  Sy,  owing  to  the  small 
deformation  of  the  bars,  we  will  also  leave  out  of  consideration. 

The  principle  of  least  work  requires  the  work  of  deformation 
of  the  truss  to  be  a  minimum,  which  means  that  the  partial  differ- 
ential coefficients  with  respect  to  the  unknowns  must  be  placed 
each  equal  to  zero.  Therefore,  we  write  the  equations  of  condi- 
tion for  our  particular  case, 

M  9M          C  N   w 

iTudx+  J-RA-W**- 

In  this  equation 

M  =  moment  with  respect  to  any  point  of  the  axis  of  any  bar, 

N  =  direct  stress  in  any  bar, 

/    =  moment  of  inertia  of  any  bar, 

A    =  sectional  area  of  any  bar, 

E   =  constant  modulus  of  elasticity, 

U  =  any  unknown. 


METHOD    OF    LEAST    WORK 


6l 


As  the  truss  is  symmetrical  in  form  and  symmetrically  loaded, 
it  suffices  to  extend  the  work  of  deformation  over  one  half  of  the 
truss,  instead  of  over  the  entire  truss. 

If  we  now  pass  a  cut  through  its  middle,  Fig.  37,  apply  the  inner 


forces  as  outer  forces  in  order  to  establish  the  equilibrium,  take  A 
as  the  origin  of  the  abscissae  x,  coincident  with  the  axis  AC  and 
of  the  abscissae  v,  coincident  with  the  axis  of  the  horizontal  bar, 
resolve  P  and  H  in  components  parallel  and  at  right  angles  to  ACt 
we  can  then  write,  H  and  M0  being  the  unknowns, 

M 
dM 


dH 


P  cos  a  X  x  —  H  sin  a  X  x  —  M0, 

dM 
.-sinaX*;  -^rr  =  -  I, 


and 


N  =  P  sin  a  +  H  cos  a, 


dH 


=  cos 


dN 
;  --—   =  o, 


dM. 


for  bar  AC. 


M  =  -  M, 


=  o; 


dM 


—  I, 


and 


N  =  H, 


W 

5       3Mn 


=  o, 


for  bar  AB. 


62  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

By  substituting  these  values  in  the  equation  of  condition,  we 
obtain,  with  respect  to  Fig.  36  and  37  two  equations  from  which 
the  two  unknowns  can  be  found. 

These  equations  are, 

CL  P  cos  a  sin  a2  ,         rL  H  sin  a2  X  x2  ,         rL  Mft  sin  a  X  x  , 

-       ~r  ~dx+       —  ~dx+       —  -dx 

Jo  -LI  Jo  •*!  Jo  2i 

s*L  P  cos  a  sin  a  ,         rL  H  cos  a2  rl  H  , 

-f  /  —  dx  +  /        — ax  +  I    —  dv  =  o 

J0  A  i  J0         ^i  Jo  ^2 

*^        '  T  7"  yj 

J0  1\  Jo  Jl  */0       ^  1  t/0    ^2 

Computing  these  integrals  and  solving  for  H  and  M0,  we  get 
H  =  P  X  j^T^          S 

(  *"  / 

and 


M0  =  P-  <"  ^  c~ 


0 

d 


where  a  =  -  -r-, 
6  /1 

i  cos  aL 


each  sin  a  has  been  replaced  by  cos  a. 
If  we  assume  7X  =  oo,  we  obtain 

H  =  o,  and  M0  =  o; 
and  if  we  assume  7\  =  o,  we  get 

H  =  P,  and  Af0  =  o, 


METHOD    OF    LEAST    WORK  63 

which  are  the  same  results  as  stated  before.     But  if  we  substitute 
the  values  as  given  in  Table  I,  we  find  the  horizontal  pull: 

H  =  124.55  tons> 
and  the  moment 

M0  =  M3  =  -  32.8  inch- tons  =  —  65,600  inch-pounds, 
M3  =  M2  =  M4  =  My    See  Fig.  39. 

M 3  has  the  same  direction  of  revolution  as  indicated  in  Fig.  37, 
and  M 2  has  the  opposite  direction  of  M 3. 
The  moment  at  the  apex  is 

Ml  =  PI  -  H  sin  a  X  L  -  32.8, 


or 


Ml  =  -f  94.47  inch-tons  =  188,940  inch-pounds. 


Fig.  38. 

This  moment  corresponds  to  a  stress  of  1020  pounds  per  square 
inch  in  the  outer  fiber,  and  is  not  more  than  8.5  per  cent  of  the 
primary  stress. 

The  stress  in  each  compression  member  is  R  =  ^124.  55* 
=  176.45  tons. 


64  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

This  stress  must  pass  through  the  point  of  inflection,  a  point 
where  no  moment  exists.  The  fact  that  the  horizontal  pull  is  less 
than  125  tons  points  to  the  displacements  of  the  directions  of  the 
compressive  stresses  as  indicated  in  Fig.  38. 

Passing  a  section  through  each  end  point  of  the  bar  AC  and 
one  through  the  point  of  inflection,  and  applying  the  stresses 
R  as  exterior  forces,  we  can  then  replace  R  by  S  =  R  cos  co  and 
Q  =  R  sin  oj,  a  moment  M^  at  point  C  and  a  moment  M2  at  A. 
The  equilibrium  requires  the  identity  in  magnitude  of  the  stresses 
5,  acting  along  the  chord  of  the  deformed  bar  and  the  transverse 
forces  Q  at  right  angles  to  this  chord  ;  further,  we  must  have 

QL2  =  Ml  and  QL,  =  M2,  or  QL  =  Ml  +  M2. 
The  location  of  the  point  of  inflection  is  found  from 


or 

LM, 


T  ,  . 

1  == 


As  /!  =   —  -1  ,  we  find  the  angle  w  from 
K 


M 


We  can  also,  as  in  Fig.  38  c,  and  without  disturbing  the  equilib- 
rium, add  at  each  of  the  points  A  and  C  two  forces,  each  equal  to 
R  in  magnitude,  but  acting  in  opposite  directions.  If  now  one 
force  R  is  resolved  in  two  components,  one  parallel  to  the  chord 
of  the  deformed  bar,  and  the  other  at  right  angles  to  it,  we  obtain 
the  same  result  as  before. 

In  order  to  test  the  accuracy  of  our  calculation,  we  apply  now 
Miiller-Breslau's  method  as  a  check. 

As  2M  =  o  with  respect  to  any  panelpoint,  and  on  account 
of  symmetry  we  must  have  in  reference  to  Fig.  39,  M±  =  M6  and 


METHOD    OF    LEAST    WORK  65 

M2  =  M3  =  M4  =  M5,    consequently    there    remain    only    two 
unknown  moments  to  be  determined. 

We  select  Mx  and  M2  as  the  two  unknowns,  and  begin  with  the 

250,Tons 


X    I 


Fig-  39- 

calculation  of  the  6  E  fold  alterations  of  the  angles  in  the  triangle 
according  to  equations  (i). 
These  values  are, 


6  £Aa  =  6  { (-  6  -  7)  i  +  (-  6  +  6)  0}  =     -  78, 
6EA/?  =  6  {(-6 +  6)0  +  (-6  +  7)i|  =     -78,- 
6  £A7  -  6  { (7  +  6)  i  +  (7  +  6)  i  j  =  =  +  156. 

The  stresses  are  given  in  net  tons  per  square  inch,  Table  I, 
consequently  the  modulus  of  elasticity  E  must  be  reduced  to  the 
net  ton  and  the  square  inch  as  the  units,  and  it  is  assumed  to  be 
14,500.  A  a,  A/9,  and  A7  are  measured  on  the  arc  for  a  radius  =  i. 

If,  for  instance,  we  suppose  that  the  bars  AC  and  BC  are  non- 
elastic,  but  the  bar  AB  elastic,  then,  according  to  Chapter  II, 


6  {7(1  +  i)} 
6  X  14500 


=  0.000965  inches; 


66  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

and  if  measured  for  a  radius  equal  to  the  height  of  the  truss, 
Fig.  40,  we  have 

h&y  =  282.84  X  0.000965  =  0.273  inches. 

The  value  0.273  inches  is  the  elongation  in  the  bar  AB,  for  the 
elongation 


oW 

Fig.  40. 

S  2  I         125   X    2   X    282.84 

A2/=  r=-r-  =  — —  — —  =0.273  inches. 

EA  14500  X  17.86 

Considering  the  identity  of  the  moments  at  the  supports  and 
apex,  we  write: 

U    — -*      -  TI    -1 ?  —  77    -*  —  U  — i  • 

1/2  L       3  /, "    4  /,      '  £ ' 

and  using  the  data  from  Table  I,  we  get 

U2  =  Us  and  U3  =  U4  =  1.885  U2 . 
With  reference  to  equations  (31)  and  (32),  we  have 

1.885*7,          -^-5-770^-78, 

U2  U,+  U2-  1.885  U2  -  78, 

Solving  for  Ui  and  C72)  we  find 

Ul  =    +  47.18,  and  U2  =  -  16.35; 
and  the  bending  moments 

M,  =  Ul  -f -  =  +  94. 36  inch-tons,  or  188,720  inch-pounds. 

JLi 

M2=  U2  —  =  -  32.70  inch-tons,  or  65,400  inch-pounds. 


METHOD  OF  LEAST  WORK  6/ 

Comparing  these  results  with  those  previously  obtained,  we 
find  the  greatest  difference  not  more  than  three  tenths  of  one  per 
cent.  The  finding  of  a  positive  Ml  means  that  the  bar  AC  is 
deformed  at  the  point  C  as  shown,  in  Fig.  39,  but  M2  being  found 
negative,  the  deformation  of  the  bar  at  the  support  is  contrary 
to  that  shown,  and  as 

M3  =  U3—2  =  U2  —  =  —  16.35  X  2=  —  32.70  inch-tons, 

the  elastic  line  of  the  bar  AB  is  also  curved  contrarily  to  what  is 
shown  in  Fig.  39.     The  deformations  are  indicated  in  Fig.  41. 


Fig.  41. 
The  displacements  of  the  lines  of  direct  stresses  are, 

/i  =  f:=5^=o-534inches- 

=  0.185  inches. 


*' /w   =  0.262   inches. 
I24-55 

The  foregoing  investigation  shows  that  the  lines  of  the  dis- 
placed stresses  form  a  diagram  which  is  not  identical  with  the 
truss  diagram,  consisting  of  the  center  lines  of  the  truss  members. 
Equilibrium  exists,  but  the  lines  of  the  stresses  do  not  intersect 
any  more  in  the  panelpoints  of  the  truss. 


CHAPTER    VIII. 

OTHER    CAUSES    OF    SECONDARY   STRESSES   THAN    RIVETED 
JOINTS    IN    MAIN   TRUSSES. 

i.     Eccentricities. 

WHEN  the  stress  in  a  truss  member  does  not  pass  through  the 
center  of  a  panelpoint,  then  its  eccentricity  not  only  brings  about 
a  change  in  the  secondary  stresses,  but  it  itself  is  a  source  of  a  new 
stress.  These  stresses  have  usually  different  signs,  which  means 
that  a  stress  due  to  an  eccentricity,  is  in  part  counterbalanced  by 
the  rigid  connection. 

In  the  execution  of  calculations,  proper  attention  must  be  paid 
to  the  character  of  the  signs  of  the  moments  due  to  an  eccentric 
connection.  So,  for  instance,  we  may  call  a  moment  positive  if  it 
turns  in  the  direction  of  the  hand  of  a  clock,  and  negative  for  a 
counter-revolution. 

If  a  bar  with  the  stress  S  has  an  eccentricity  at  each  end  c  and  cit 
then  M  =  Sc  and  Mt  =  Scly  and  the  angle  of  deflection  -c  on 
account  of  the  eccentricity  is,  according  to  equations  (2), 


C  6  El  6  El 

The  angles  between  the  different  bars  of  a  truss  are  now  sub- 
jected to  a  change,  not  only  owing  to  alterations  in  their  lengths, 
but  also  on  account  of  eccentricities;  and  after  having  taken  care 
of  the  deflection  angles  TC  with  their  proper  signs  in  the  determi- 
nation of  the  EAce,  the  calculations  proceed  then  as  previously 
explained.  After  the  calculations  are  finished,  the  direct  effects 
of  the  eccentric  connections  must  be  added  to  the  secondary 

stresses. 

68 


OTHER    CAUSES    OF    SECONDARY    STRESSES 


69 


2.    Loads  between  Panelpoints  in  the  Plane  of  the  Truss. 

These  loads  are  dead  and  live  load  and  braking  forces.  The 
live  load  between  panelpoints  could  be  avoided  by  designing  a 
bridge  with  floorbeams  and  stringers;  and  in  case  a  floor  is  riveted 
to  the  posts  at  any  point  between  top  and  bottom  of  the  main 
trusses  so  that  the  posts  are  subjected  to  bending  by  the  braking 
of  trains,  the  insertion  of  extra  members  will  transmit  the  braking 
forces  to  the  panelpoints  without  causing  bending  in  the  posts. 

The  calculation  of  the  effects  of  loads  applied  between  panel- 
points  is  about  the  same  as  that  for  eccentricities.  If  we  have 
to  deal  with  dead  load,  for  instance,  we  consider  each  member 
individually,  calculate  the  stress  SM  due  to  its  own  weight,  deter- 
mine further  the  deflection  angle  r  at  both  ends  caused  by  it,  and 
proceed  then  similarly  as  shown  for  eccentricities.  Finally,  we 
add  the  stresses  s^  and  the  secondary  stresses. 

3.  Loads  between  and  at  the  Panelpoints  of  a  Member  supposed 
to  turn  freely  around  a  Pin. 

Under  this  head  comes  an  eyebar  whose  secondary  stresses  are 
of  particular  interest,  as  has  been  shown  by  the  discussion  they 
caused  among  engineers.  The  exact  solution  of  the  problem 


requires  the  simultaneous  consideration  of  the  action  of  its  dead 
weight,  which  consists  in  a  deflection,  and  that  of  a  pull  reducing 
in  part  this  deflection.  A  separate  consideration  of  dead  weight 
and  pull  leads  to  approximate  results.  We  assume  an  eyebar 


70  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

under  tension,  the  centers  of  the  two  heads  in  a  horizontal  line 
and  one  half  of  the  bar  walled  in,  which  circumstance  will  not 
in  any  way  disturb  the  equilibrium.  Further,  we  call  P2  the  pull, 
Pt  the  reaction  from  the  dead  weight  of  the  bar,  supposed  to  be 
uniformly  distributed  over  its  length,  and  take  A  as  the  origin  of 
coordinates,  Fig.  42. 

The  problem  consists  now  in  the  representation  of  the  ordinate 
y  as  a  function  of  oc.  The  relation  between  these  two  variables 
is  the  equation  of  the  elastic  line;  and  as  soon  as  the  latter  is  found, 
it  is  easy  to  calculate  the  deflection,  the  moment,  and  the  stress 
for  any  section  of  the  bar. 

With  respect  to  Fig.  42  the  equation  of  the  elastic  line  is  written, 


ii> 

dx2  2  L 

since  the  negative  ordinates  are  below  the  axis  of  the  abscissae. 

P  P 

Calling  the  constants    —^-  =  n  and  ~  =  q, 
El  ILL 

(Py          (  x2  >   , 

we  have  ^  =  n  j  -  *  +  -  j  +  qy. 

To  facilitate  the  investigation,  we  write  /  instead  of  -f-  and 

doc 

d?y 

instead  of   -r^2  ,  and  our  given  equation  becomes  now 

nx2 

y"  =  qy  -  nx  +  — 

Differentiating  twice,  we  obtain 

£-»•-+"• 


By  integrating  this  last  equation  twice,  we  obtain  /'  as  a  func- 
tion of  x\  and  if  this  function  of  x  is  substituted  in  the  given 


OTHER    CAUSES    OF    SECONDARY    STRESSES  71 

equation,  we  find  then  at  once  y  as  a  function  of  x,  which  is  the 
object  sought. 

Multiplying  the  last  equation  with  dy",  we  have 

dy"  X  9r  -  d-f  X  d\  *f  I   =  gyW  +  5  rf/. 
dx2        doc  I  dx   )  L   ' 

/ ) 2 

Since  Xd  is  the  differential  of       dx  '    , 

dx  (  dx  )  2 


therefore  -  is  the  integral  of  -f-    X  d 

2  dx 

and  we  get  by  integration, 

!£!'-*    /W  +  **fdf  +  A-  &•  +    I*    y"  +  A, 


or 

^y  j 

=  =   0#. 


Integrating  again,  we  obtain 
i         (      n 


A  and  5  are  arbitrary  constants;  and  as  these  can  be  changed, 
we  write  the  last  equation, 


--   "  +  A 


B 

or 


_  +  y/V0  +  \/  ^}'//2  +  —  y"  + 
~B~ 


?2  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

£  is  the  base  of  the  system  of  natural  logarithms.     From  this 
equation  we  obtain  the  value  of  y"  as  a  function  of  x,  and  by  sub- 

nx2 

stitution  of  this  value  in  the  given  equation  y"  =  qy  —  nx  +  -    -  , 

2  L 

we  get  after  some  simple  transformations, 

A  _,y-       n    {  nx     nx* 


^ 
2  qVq       I  2  L2q2V/q  XB     2  q\/q  X  B  )  L(f       q       2  Lq 

The  next  step  consists  in  the  determination  of  the  constants. 
Putting 


2  L2q2Vq  XB       2  qVq  XB  2  qVq 

C  and  D  being  two  ne\v  constants,  \ve  have 


_ 
2 


Lq2        q        2  Lq 
and  by  differentiation, 


dx        q      Lq 

dy 
From  the  conditions  that  y  =  o  for  x  =  o,  and  ~  =  o  for  x  =  L, 

the  two  constants  C  and  D  are  found. 
They  are 

C_ __^___ 
t    ^  */  ~            r+i~\y 


and  if  C  and  D  in  the  equation  for  y  are  replaced  by  these  values, 
then 


OTHER    CAUSES    OF    SECONDARY    STRESSES  73 

EXAMPLE.  An  eyebar  15  inches  wide  by  2  inches  thick  and 
55  feet  long  is  subjected  to  a  pull  P2  =  600,000  pounds,  or  20,000 
pounds  per  square  inch.  Required  the  maximum  bending  stress. 

Let  the  modulus  of  elasticity  be  29,000,000  pounds  per  square 
inch, 

Pl  =  2800  pounds, 

/  =562.5, 

L    ••=  330  inches, 


The  maximum  deflection  D  is  equal  to 


D =- < -  +  - 
q  (  2        Lq 


-  i  j  >  =  o.  488  inches. 


The  maximum  bending  moment  is  equal  to 

M  =     -  2800  X  330  +  2800  X  165  +  0.488  X  600,000, 
or      M  —  —  169,200  inch-pounds,  consequently    the    maximum 
bending  stress  equals  -  -  =  2256  pounds  per  square   inch. 

This  bending  stress  amounts  to  about  11.3  per  cent  of  the 
direct  stress,  and  is  compression  in  the  top  fiber  and  tension  in 
the  bottom  fiber. 

4.    Changes  in  Temperature. 

Any  rise  or  fall  in  temperature  affects  the  lengths  of  truss  mem- 
bers. These  alterations  in  the  lengths  of  bars  produce  deforma- 
tions of  a  truss,  which  may  or  may  not  be  connected  with  stresses. 

A  statically  determinate  truss  whose  movable  end  is  free  from 
any  frictional  resistances  and  whose  members  can  turn  freely 
around  pins,  is  not  subjected  to  any  temperature  stresses,  not  even 
when  its  members  are  unequally  affected  by  temperature. 

But  the  case  is  different  with  statically  indeterminate  trusses, 
no  matter  whether  the  indeterminateness  refers  to  the  outer  or 


74  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

inner  forces.  The  safety  of  the  structure  requires  an  examination 
of  its  temperature  stresses,  and,  if  necessary,  an  inquiry  into  the 
secondary  stresses  arising  from  them.  Statically  determinate 
trusses,  which  are  riveted,  can  also  be  affected  by  temperature 
stresses.  So,  for  instance,  can  a  top  chord  have  a  higher  tempera- 
ture than  a  bottom  chord,  if  the  latter  is  protected  from  the  rays 
of  the  sun  by  a  floor.  In  this  case  the  difference  in  temperature 
produces  an  upward  deflection  of  the  truss,  and  consequently 
deformations  and  stresses  in  the  bars. 

The  calculation  of  temperature  stresses  is  as  follows: 
If  c  is  the  coefficient  for  expansion  or  contraction  due  to  i° 
change  in  temperature,  and  /  the  total  change  of  temperature  in 
degrees,  then 

Sl         >i 
EA  ^  C»> 
or 


A 

This  value  of  st  is  now  used,  —  for  instance,  -in  equations  (i), 
-  in  order  to  calculate  the  angle  alterations,  whereupon  the  suc- 
ceeding operations  are  executed  as  formerly  explained. 

5.    Misfits. 

It  is  very  essential  that  the  lengths  of  members  for  a  riveted 
truss  are  exact;  if  they  are  not,  they  are  a  source  of  secondary 
stresses.  We  speak  here  in  particular  of  statically  indeterminate 
trusses  where  a  small  shop  mistake  may  lead  to  a  considerable 
change  in  the  stresses.  In  order  to  determine  the  effects  of  these 
misfits  on  the  stresses,  we  assume  that  these  misfits  are  produced 
by  stresses  sm,  and  if  we  call  A/  the  amount  a  truss  member  is 
either  too  long  or  too  short,  we  can  write 


or 

sm=—  X  E. 


OTHER    CAUSES    OF    SECONDARY    STRESSES  75 

This  unit  stress  sm  is  now  used  in  the  same  way  as  the  unit 
stress  st  due  to  a  change  in  temperature. 

6.    Brackets  on  Posts. 

Eccentric  loads  on  posts  caused  by  brackets  give  rise  to  very 
lengthy  calculations,  and  as  these  loads  affect  the  entire  cross- 
frame  of  a  bridge  we  cannot  consider  them  here.  It  is  best  to 
avoid  such  brackets  wherever  possible. 

7.    Unsymmetrical  Connections. 

Good  practice  does  not  allow  unsymmetrical  connections  in 
the  design  of  main  trusses;  and  at  such  places  where  they  are 
tolerated,  the  secondary  stresses  caused  by  them  are  of  minor 
importance. 

8.    Curved  Members. 

The  secondary  stresses  due  to  curved  members  can  be  computed 
from  the  suggestions  given  for  eccentric  connections.  Bridge 
trusses  with  curved  members  do  exist,  but,  in  the  opinion  of  the 
writer,  they  are  utterly  out  of  their  proper  place.  The  defense 
of  such  members,  even  from  an  aesthetic  point  of  view,  is  weak. 

9.    Pin  Joints. 

We  believe  it  to  be  a  safe  statement  that  many  an  engineer  lays 
too  great  a  stress  on  the  value  of  a  pin  joint  as  a  means  to 
reduce  secondary  stresses.  Of  course  no  pin  joint  is  perfect,  and 
in  some  cases  the  frictional  resistance  may  be  so  small  that  we  do 
not  need  to  pay  any  attention  to  it.  On  the  other  hand,  if  a  pin 
is  designed  with  no  consideration  whatever  for  a  reduction  of 
secondary  stresses,  there  is  at  least  a  chance  that  it  will  be  in- 
effective, so  that  a  riveted  joint  could  just  as  well  have  been  built. 

If  a  bar  does  turn  around  a  pin,  it  is  certain  that  the  stress  in 
the  bar  will  be  displaced  out  of  its  former  axial  position,  Fig.  43. 
In  this  case  the  displacement  r  is  such  that  the  moment  Sr  over- 


76  SECONDARY    STRESSES     IN    BRIDGE    TRUSSES 

comes  the  frictional  moment  F  X  R,  F  being  the  frictional  resist- 
ance and  R  the  radius  of  the  pin. 

The  frictional  resistance  is  found  by  resolving  the  stress  S  at 
its  point  of  application  on  the  pin  periphery  into  two  components, 
the  line  of  one  component  coinciding  with  the  tangent  on  the  pin 
periphery,  and  the  other  normal  to  this  tangent.  The  frictional 


Fig-  43- 

resistance  is  now  equal  to  the  tangential  stress  5  X  sin  0  and  also 
equal  to  the  normal  pressure  S  X  cos  <£  multiplied  by  the  coeffi- 
cient of  friction.     The  angle  $  included  between  the  line  of  the 
stress  S  and  the  normal  is  the  angle  of  friction. 
We  have  then, 

FxR  =  SXsin(t>xR  =  SXr, 

and  r  =  R  X  sin  <f>. 

.  This  shows  the  displacement  r  is  independent  of  the  stress  S. 
If  this  displacement  is  smaller  than  r,  then  no  turning  of  the  bar 
around  the  pin  is  possible ;  and  if  it  equals  r,  a  turning  takes  place. 
Let  us  now  go  back  to  the  triangular  riveted  truss,  calculated 
in  the  preceding  chapter,  in  order  to  find  for  a  given  coefficient 
of  friction  the  greatest  diameter  of  the  pin  at  the  apex,  which 
must  not  be  exceeded  if  a  turning  of  the  bars  around  the  pin  is  to 
be  realized.  As  the  displacement  of  the  stress  in  this  case  was 


OTHER    CAUSES    OF    SECONDARY    STRESSES  77 

found  =  0.534  inches,  and  assuming  0.2  as  the  value  of  the 
coefficient  of  friction,  we  have  0.534  =  0.2  R,  or  R  =  2.67  inches, 
which  means  that  if  we  design  this  pin  with  a  greater  diameter 
than  2  X  2.67  =  5.34  inches,  it  would  be  ineffective,  and  with  a 
greater  frictional  coefficient  the  diameter  of  the  pin  should  of 
course  be  still  smaller. 

From  the  above  considerations  it  follows  that  if  we  wish  to 
reduce  the  secondary  stresses  to  a  minimum  in  hinged  members, 
it  is  very  essential  to  keep  the  size  of  the  pins  down  as  much  as  the 
strength  and  safety  of  a  structure  will  permit. 

The  writer  is  not  aware  that  experiments  have  been  made  with 
a  view  to  determine  the  frictional  coefficient  for  cases  that  are  here 
under  consideration,  and  therefore  he  is  not  in  a  position  to  fur- 
nish any  reliable  data. 

Some  specifications  require  that  the  diameter  of  a  pin  shall  not 
be  less  than  three  quarters  of  the  width  of  any  eyebar  which  they 
connect.  Calling  S  the  direct  stress  of  an  eyebar,  w  and  /  its 
width  and  thickness,  0.2  the  frictional  coefficient,  then  we  have 

for  the  displacement  of  the  stress,  r  =  0.375  w  X  0.2  =  0.075  w- 

$ 

The  direct  unit  stress  equals  • —  ,  and  the  bending  stress  is  equal 

wt 

to  the  moment  divided  by  the  section  modulus,  or  equal  to 

S  X  0.075  X  w  X  6  =  S_ 
inH  wt 

§ 

The  sum  of  the  direct  and  bending  stress  equals  —  X  } i  +  0.45}'. 

which  means  that  the  secondary  stress  amounts  to  45  per  cent  of 
the  primary  stress. 

In  regard  to  these  stresses  it  should  be  noted  that  probably  the 
vibrations  due  to  a  passing  train  cause  the  eyebars  to  adjust  them- 
selves to  their  original  positions;  they  probably  turn  around  their 
pins,  whereby  the  angle  of  friction  is  momentarily  decreased  from 
what  it  would  be  for  static  loads. 


78  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

10.     Friction  at  Supports. 

The  frictional  resistances  at  the  supports  of  trusses  are  depend- 
ent on  the  coefficient  of  friction,  the  vertical  loads,  and  the  length 
of  span.  These  resistances  would  -be  very  great  indeed  for  long 
trusses,  if  sliding  friction  were  allowed;  but  as  for  such  trusses 
only  rolling  friction  is  allowed,  the  resistances  are  very  con- 
siderably reduced,  and  need  not  to  be  considered.  But  it  is,  of 
course,  important  that  the  roller  ends  be  kept  in  proper  working 
order. 

ii.     Cross-Frames. 

The  analytical  discussion  of  cross-frames,  if  a  complete  solution 
of  the  problem  is  attempted,  leads  to  very  extensive  investigations 
and  is  outside  of  our  province.  Nevertheless,  we  will  give  a  few 
remarks. 

A  thorough  examination  of  cross-frames  for  either  a  through  or 
deck  railway  bridge  consisting  of  verticals,  floorbeams,  and  cross- 
constructions  of  any  description,  would  consider  the  bending 
effects  on  the  frame  of  the  following  forces:  dead  load  of  floor- 
beams,  dead  and  live  loads  transferred  from  the  stringers  to  the 
floorbeam,  centrifugal  force,  impact,  wind  pressure  against  the 
train,  wind  pressure  against  the  structure  concentrated  at  top 
and  bottom  and  uniformly  distributed  against  the  posts,  unequal 
deflection  of  the  main  trusses,  and  unequal  change  of  temperature 
for  different  parts  of  the  frame. 

The  influence  of  these  forces  is  felt  in  the  entire  cross-frame, 
causing  also  bending  and  twisting  in  the  members  of  the  main 
trusses. 

The  secondary  stresses  in  cross-frames  are  next  in  importance 
to  those  in  the  main  trusses ;  and  concerning  cross-frames  with  no 
diagonals,  it  may  be  said  that  they  are  predisposed  to  higher 
stresses. 

For  the  purpose  of  finding  out  to  what  extent  the  secondary 
stresses  are  affected  in  the  verticals  by  dead  and  live  load,  and 
exclusively  by  changing  the  dimensions  of  the  verticals,  the  writer 


OTHER    CAUSES    OF    SECONDARY    STRESSES  79 

examined  a  riveted  cross-frame  of  a  two-track  railway  bridge, 
consisting  of  a  6-foot-deep  floorbeam,  a  lattice  strut,  and  two  sus- 
penders, each  of  the  latter  composed  of  4-8-inch  bulb  angles  with 
a  total  area  of  22.5  square  inches. 

Deep  floorbeams  with  large  moments  of  inertia  tend  toward  a 
reduction  of  secondary  stresses. 

The  trusses  were  30  feet  from  center  to  center,  the  panel  length 
27  feet,  and  the  live  load  carried  by  each  of  the  suspenders  187,000 
pounds. 

The  lengths  of  the  verticals  and  their  depths  parallel  to  the 
web  of  the  floorbeam  were  the  only  dimensions  changed;  and  if 
we  express  the  secondary  stresses  in  percentages  of  the  unit  stress 
due  to  dead  +  live  load,  the  results  are  as  follows: 

Per  cent. 

Verticals  36  feet  long  and  14^  inches  deep I3-S 

Verticals  36  feet  long  and  29    inches  deep 23-3 

Verticals  18  feet  long  and  14^  inches  deep 25.2 

Verticals  1 8  feet  long  and  29    inches  deep 40 .9 

A  cross-frame  as  described  36  feet  deep  may  belong  to  a  through 
Warren  truss  bridge,  the  verticals  being  suspenders;  and  a  cross- 
frame  1 8  feet  deep  may  belong  to  a  Baltimore  deck  bridge  with 
the  top  chord  projecting  above  the  floor,  the  verticals  being  short 
posts.  For  the  analytical  treatment  it  does  not  make  any  differ- 
ence whether  the  floorbeam  is  at  the  bottom  or  top. 

In  another  example,  the  writer  selected  a  cross-frame  of  a  four- 
track  railway  through  bridge,  with  two  main  trusses,  symmetri- 
cally loaded,  gave  purposely  the  post  the  unusual  transverse  depth 
of  52  inches,  designed  it  under  three  prominent  specifications,  and 
found  in  each  case  that  the  secondary  stress  amounted  to  closely 
55  per  cent  of  the  primary  stress.  But  by  reducing  the  depth 
parallel  to  the  floorbeam  of  this  comparatively  short  post  to  18 
inches  with  ample  provision  against  buckling,  the  secondary  stress 
was  reduced  to  22.3  per  cent. 

These  high  stresses  in  the  verticals  are  verified  by  Winkler, 
Querkonstruktionen,  pp.  179-182;  by  Jebens,  fiie  Spannungen  in 


8o  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

den  V  erticalstandern  der  eisernen  Br'ucken,  Zeitschrijt  des  Vereins 
deutscher  Ingenieure,  1880,  p.  127;  and  by  that  standard  work 
of  engineering  science,  Handbuch  der  Ingenieurwissenschaften, 
vol.  II. 

From  the  above  we  can  draw  the  conclusion  that  a  small  ratio 
between  the  length  and  the  transverse  depth  of  a  vertical  should 
be  avoided,  and  that  the  transverse  depth  should  be  restricted  to 
a  proper  limit,  otherwise  we  run  the  risk  of  exceeding  by  far  the 
unit 'stresses  as  prescribed  by  our  specifications. 

It  is  of  interest  to  note  that  investigations  of  riveted  cross-frames 
disclose  the  assumption  of  the  fixity  of  floorbeams  at  the  posts  as 
quite  erroneous;  the  floorbeams  cannot  even  be  approximately  so 
considered. 

12.    Yielding  of  Foundations  and  Settlements  of  Masonry. 

The  influence  on  the  stresses  of  a  truss  caused  by  yielding  of 
the  foundations  or  settlements  of  masonry  could  be  calculated 
if  these  displacements  were  known.  As  a  matter  of  fact,  they 
can  only  be  estimated  or  judged  from  uncertain  or  incomplete 
evidence. 

In  regard  to  these  displacements  we  must  distinguish  between 
elastic  and  non-elastic  deformations.  Elastic  deformations  vanish 
as  soon  as  the  load  is  removed,  while  the  non-elastic  are  perma- 
nent. 

A  structure  should  not  be  built  if  the  computations  prove  that 
it  is  very  sensitive  to  assumed  displacements  of  its  supports,  pro- 
vided we  cannot  give  it  supports  which  are  almost  as  good  as  fixed. 
It  is  also  essential  in  such  a  case  that  the  supports  are  placed  with 
the  utmost  care  in  those  positions  as  assumed  in  the  computations. 

Of  course  not  every  truss  is  affected  by  a  displacement  of  its 
supports.  So,  for  instance,  a  truss,  resting  on  two  supports  and 
with  one  movable  end,  a  cantilever  truss  or  a  three-hinged  arch 
are  free  from  this  influence.  On  the  other  hand,  a  continuous 
truss  is  very  susceptible  to  the  influence  of  a  yielding  of  the  founda- 
tions or  a  settlement  of  the  piers  (particularly  so  if  very  massive 


OTHER    CAUSES    OF    SECONDARY    STRESSES 


8l 


and  deep);  also  the  one-hinged,  the  two-hinged,  and  the  hingeless 
arch  and  others  are  affected  by  these  causes. 

Continuous  truss  over  three  supports.  We  will  suppose  that  the 
center  pier  of  a  continuous  truss  over  three  supports  yields  verti- 
cally to  the  amount  of  D  inches,  Fig.  44.  Such  a  displacement 
reduces  the  center  reaction  an  amount  X,  which,  when  found, 
enables  us  to  compute  the  stresses  in  the  truss  members,  and  con- 
sequently also  the  secondary  stresses.  In  order  to  find  X,  we 


place  a  load  equal  to  unity  at  the  center  of  the  truss,  the  latter 
assumed  to  be  resting  on  its  end  supports  only,  and  determine 
the  deflection  Dl  at  the  center  due  to  this  load  =  i.  If  we 
now  apply  a  load  X  at  the  center  of  the  truss,  then  the  deflection 
will  be  XD^  but  as  this  deflection  must  be  equal  to  the  supposed 
deflection  D,  we  have 

XDl  =D,orX  ==— - 

The  examination  of  a  continuous  plate  girder  is  quite  similar 
to  that  of  a  truss. 

Two-hinged  arch.  Of  all  forms  of  arches,  the  two-hinged  arch 
has  probably  found  the  widest  application,  and  from  this  reason 
we  will  take  it  as  an  example.  This  kind  of  arch  is  statically 
indeterminate  with  one  unknown  quantity,  which  is  the  hori- 
zontal thrust,  provided  the  truss  has  no  redundant  members. 

The  principle  of  the  derivative  of  work  furnishes  us  with 
convenient  means  to  determine  the  thrust.  For  this  purpose  we 
first  remove  the  statical  indeterminateness  by  giving  the  truss 
one  movable  end,  the  two  supports  being  supposed  in  this  case  in 


82  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

a  horizontal  line.  The  stress  S0  in  any  of  the  truss  members, 
due  to  vertical  loading,  can  now  be  found  by  statics.  There- 
upon we  apply  a  force  =  i  at  the  movable  end,  acting  toward 
the  fixed  hinge,  and  determine  the  stress  v  either  analytically  or 
graphically  in  every  member  of  the  truss.  If  now  the  real  hori- 
zontal thrust  =  H,  and  if  5  denotes  the  stress  in  any  of  the  bars 
of  the  statically  indeterminate  truss,  we  have 

5  =  50  +  vH. 

The  stress  S0  is  a  function  of  the  exterior  forces,  and  is  independ- 
ent as  well  as  the  stress  v  of  the  thrust  H.  The  work  of  deforma- 
tion W  is  expressed  by 

^-y^l 

M    -*aEA' 

where  I  =  length  of  any  bar, 

A  =  sectional  area  of  any  bar, 
E  =  modulus  of  elasticity. 

The  principle  of  the  derivative  of  work  states  that,  if  we  express 
the  work  of  deformation  of  the  bars  as  a  function  of  the  exterior 
forces,  then  the  displacement  of  the  point  of  application  of  a  force 
(in  our  case  H)  equals  the  partial  derivative  of  the  work  of  defor- 
mation with  respect  to  that  force. 

Differentiating  the  work  of  deformation  with  respect  to  H}  we 
get 

ITT    ~      '     '  r>    A     ^      ITT  ~~~      ^*<*J 


L  being  the  length  of  the  span  from  center  to  center  of  end  pins, 
and  AL  the  supposed  horizontal  outward  yielding  of  the  masonry 
supports  under  the  action  of  the  vertical  loading. 

In  case  temperature  stresses  are  to  be  considered,  wre  call  the 

SI  SI 

alteration  in  the  length  of  a  bar:——  +  etl  instead  of  -=rr;  £  being 

EA  k,A 

the  coefficient  for  extension  or  contraction  due  to  a  change  in 
temperature  of  i  degree  and  for  a  unit  of  length,  and  /  the  total 
change  in  temperature. 


OTHER    CAUSES    OF    SECONDARY    STRESSES  83 

But  6"  =  S0  +  vH>  and  — -  =  ^,   consequently   we    obtain    by, 
substitution, 


or 

TT 


The  derivation  of  this  formula  for  H  assumes  that  no  initial 
stresses  exist,  that  is  to  say,  with  the  removal  of  the  outer  loading 
all  stresses  must  vanish. 

The  three  different  causes  which  influence  the  thrust  of  the 
arch  may  also  be  considered  separately. 

The  thrust  caused  by  the  sole  action  of  the  vertical  load  is 


and  by  a  change  in  temperature, 

X  v 


v2l 


[t  being  positive  for  an  increase]  and  by  a  change  in  the  length 
from  center  to  center  of  end  hinges, 


being  positive  for  an  increase]. 
The  expressions  ^TrV  X  v  and  ^TTT~  designate  horizontal 

*  zZ/^T.  jC/.ri. 

displacements  of  the  hinges,  which  are  supposed  to  move  freely; 
the  former  is  the  displacement  caused  by  the  vertical  load,  and 
the  latter  is  that  due  to  a  horizontal  thrust  equal  to  unity  and 
applied  at  the  hinge. 


84  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

The  length  of  the  distance  between  the  hinges  can  be  increased 
by  the  thrust  of  the  arch  in  pushing  the  supports  bodily  outward 
or  by  crushing  or  compressing  the  masonry.  Such  an  action 
naturally  decreases  the  horizontal  thrust  and  consequently  exer- 
cises an  influence  on  the  stresses  in  every  member  of  the  arch 
truss  with  corresponding  changes  in  the  secondary  stresses. 

As  an  example  of  a  thorough  examination  of  the  effects  of  the 
yielding  of  masonry  supports  on  the  stresses  in  a  truss,  can  be 
mentioned  the  bridge  across  the  Emperor  William  canal  at  Gruen- 
enthal,  described  by  Fiilscher  in  "  Zeitschrift  fiir  Bauwesen," 
1898.  This  bridge  is  built  for  a  single-track  railway  and  highway 
traffic,  has  two  crescent-shaped  arch  ribs,  projecting  above  the 
floor  system,  and  measures  513.3  feet  from  center  to  center  of  end 
pins. 

The  wind  bracing  in  the  plane  of  the  floor  consists  of  a  wind 
chord  and  diagonals,  and  the  place  of  the  struts  is  taken  by  the 
floorbeams.  It  is  divided  into  three  sections:  a  central  section, 
lying  between  the  intersection  points  of  the  arch  ribs  and  the  floor; 
and  two  end  sections,  each  extending  from  these  intersection 
points  to  the  abutments.  The  central  section  of  the  bracing 
transfers  the  wind  pressure  to  the  arch,  and  the  end  sections  partly 
to  the  arch  ribs  and  partly  to  the  abutments. 

In  the  central  section  the  floor  is  suspended,  and  in  the  end 
sections  it  is  supported  by  posts  resting  on  the  arch  ribs. 

Apart  from  the  effects  of  the  dead  load,  live  load,  wind  pressure, 
and  temperature  changes,  the  stresses  in  each  truss  member  have 
been  calculated  under  the  supposition  that,  before  riveting  up 
the  wind  chords,  but  after  the  arch  carried  its  own  weight,  a  hori- 
zontal yielding  of  each  of  the  masonry  supports  of  it  of  an 
inch  would  take  place,  and  a  further  yielding  of  the  supports 
of  iV  of  an  inch  under  the  influence  of  the  live  load. 

While  the  two-hinged  arch  is  affected  only  by  horizontal  dis- 
placements of  the  masonry,  the  one-hinged  and  hingeless  arch  are 
susceptible  to  horizontal  and  vertical  displacements,  and,  moreover, 
to  a  possible  turning  of  the  masonry  in  the  plane  of  the  truss. 


CHAPTER  IX. 
IMPACT. 

IT  is  outside  the  scope  of  this  book  to  take  the  reader  over  the 
field  of  mathematical  investigations  of  dynamical  effects  on  bridges, 
or  to  discuss  the  many  suggestions  that  have  been  made,  how 
impact  and  vibrations  could  be  covered  in  our  specifications.  If 
it  had  been  the  intention,  it  would  have  been  proper  to  extend  the 
discussion  of  secondary  stresses  under  static  loads  to  the  cross- 
sections  of  bridges,  floors,  and  wind  bracings  first,  before  giving 
some  notes  on  the  secondary  stresses  under  moving  loads.  But 
a  few  words  on  this  subject  are  not  out  of  place. 

While  the  progress  in  the  theory  of  bridges  has  been  gigantic, 
the  same  cannot  be  said  of  the  theory  of  dynamical  effects  on 
bridges,  and  the  reason  for  this  is  not  far  to  seek. 

If  every  element  had  to  be  considered  which  has  some  connection 
with  the  effect  produced  on  a  bridge  by  a  fast-moving  train,  then 
the  problem  of  impact  would,  of  course,  be  insoluble,  and  even  in 
the  simplest  case.  But  barring  such  elements,  as,  for  instance,  a 
defective  track,  or  inequalities  of  the  rail  ends  at  rail  splices,  etc., 
whose  influence  is  naturally  outside  the  province  of  a  calculation, 
the  mathematical  difficulties  presented  by  the  problem  are  almost 
unsurmountable ;  and,  in  fact,  they  have  been  only  overcome  for 
the  case  of  a  single  load  moving  over  a  beam. 

A  train  in  passing  over  a  bridge  causes  the  latter  to  deflect, 
whereby  the  pressure  or  centrifugal  force  exerted  by  the  train 
against  the  bridge  is  influenced  by  the  deflection  and  the  velocity 
of  the  moving  masses,  and  this  pressure  in  turn  exercises  an  influ- 
ence on  the  form  of  the  deflection  curve.  The  mutual  relations 
between  the  quantities  which  enter  into  consideration  are  compli- 

85 


86  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

cated,  and  made  more  so  by  the  counterweights  of  the  locomotive 
drivers,  which  affect  the  values  of  the  pressures.  In  consequence 
of  the  great  velocity  with  which  a  train  enters  a  bridge,  of  the 
variable  loads  produced  by  the  counterweights  of  the  locomotive, 
of  defective  rail  splices  and  unround  wheels,  the  bridge  is  subjected 
to  vibrations.  Not  only  does  the  bridge  as  a  whole  vibrate  ver- 
tically and  horizontally,  but  also  the  different  bars  perform  rapid 
oscillations  longitudinally  and  transversely. 

The  view  held  by  some  engineers,  that  a  fast-moving  train  does 
not  give  a  truss  the  necessary  time  to  offer  its  full  resisting 
power,  does  not  harmonize  with  the  fact  that  the  propagation  of 
stresses  in  elastic  bodies  follows  the  laws  governing  the  velocity  of 
sound.  The  velocities  of  sound  vary  greatly  in  different  mediums; 
in  liquids  the  velocity  is  greater  than  in  air,  and  in  solids  the  range 
is  rather  wide.  In  caoutchouc  the  velocity  is  from  100  feet  to  200 
feet  per  second,  while  in  steel  wire,  wrought  iron,  and  steel  it 
amounts  in  round  figures  to  16,000  feet,  or  about  3  miles  per  second. 
A  telegraph  wire  furnishes  a  good  illustration  of  the  propagation 
of  sound  in  solids.  Filing  at  one  end  of  the  wire  can  be  heard  at 
a  distance  of  several  miles  by  placing  the  other  end  in  the  ear. 

The  subject  in  question  has  been  treated  from  various  points 
of  view.  In  the  year  1890,  Professor  Ritter,  of  Aachen,  published 
calculations  as  a  result  of  theoretical  considerations 'which  place 
the  velocity  of  the  propagation  of  impulses  in  wrought  iron  as 
high  as  three  miles  per  second,  and  Professor  Mach  showed 
optically  the  propagation  of  longitudinal  vibrations.  Professor 
Radinger  treats  the  subject  in  his  book  on  steam  engines  with  high 
piston  velocities,  published  in  Vienna,  1892.  In  the  same  year 
appeared  in  "  Zeitschrift  des  osterreichischen  Ingenieur-  und 
Architekten-Vereins,"  a  paper  of  great  interest  to  engineers  on 
"  Metal  Constructions  of  the  Future"  by  the  late  Professor  Steiner. 

In  this  paper  Steiner  shows  how  impulses  can  be  made  visible 
to  the  eye  by  the  construction  of  a  model  of  a  bridge  truss,  each 
bar  of  the  truss  to  be  provided  with  a  groove.  If  such  a  model  is 
placed  in  a  horizontal  position  and  the  grooves  filled  with  quick- 


IMPACT  87 

silver,  it  is  then  possible  to  follow  the  propagation  of  an  impulse, 
which  has  been  made  by  the  finger  at  any  point  of  the  truss. 

However,  it  should  be  remarked  that  the  propagation  of  stresses 
from  section  to  section  in  the  members  of  a  bridge  truss  experiences 
a  delay,  firstly,  because  the  line  of  progress  must  be  changed,  and 
secondly,  on  account  of  the  imperfections  of  the  joints.  A  pin- 
connected  truss  appears  to  be  at  a  disadvantage  compared  with 
a  riveted  steel  truss,  as  the  latter  resembles  more  closely  a  con- 
tinuous mass.  But,  whatever  may  be  left  of  the  velocity  of  propa- 
gation of  stresses,  it  appears  to  be  of  sufficient  magnitude  to  be 
looked  upon  as  instantaneous. 

Under  the  assumption  that  the  stresses,  in  traversing  a  truss, 
encounter  so  many  difficulties  which  reduce  their  velocity,  Radinger 
arrives  at  the  conclusion  that  a  truss  may  be  subjected  at  the  ends 
to  particular  high  stresses  by  fast-running  trains.  He  arrives  at 
this  conclusion  in  this  way:  Let  us  suppose  for  an  illustration 
that  the  velocity  of  stresses  is  reduced  from  16,000  feet  per  second 
to  4000  feet  per  second,  then  the  time  required  by  a  span  of  400 
feet  in  length  to  act  as  a  structure  on  two  supports  would  be  2  X 

400       i      ,. 

-  =  —   or  a  second.     A  train  running  over  the  bridge  with  a 
4000      5 

speed  of  90  feet  per  second  would  cover  a  length  of  18  feet  in  ^ 
of  a  second,  which  means  that  for  this  time  the  bridge  would  have 
only  one  support  for  the  live  load. 

The  solution  of  the  problem  of  impact,  but  only  in  the  simplest 
case,  namely  for  a  single  constant  load,  moving  over  a  weightless 
beam  of  uniform  sectional  area  and  resting  on  two  rigid  supports, 
has  been  rigidly  effected  by  Dr.  H.  Zimmermann,  whose  brilliant 
researches  are  contained  in  his  paper,  "Die  Schwingungen  Eines 
Tragers  Mit  Bewegter  Last,"  Berlin,  1896.  Many  investigators 
have  attacked  the  problem  without  success  on  account  of  the 
mathematical  difficulties.  But  in  this  respect  it  should  be  remarked 
that  the  general  integral  of  the  differential  equation  of  the  curve 
described  by  the  moving  mass  is  of  a  kind  that  was  unknown  up 
to  the  publication  of  Zimmermann's  writing,  although  he  knew  it 


88  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

as  early  as  1892.  Zimmermann  tells  us  that  he  hesitated  to  pub- 
lish the  purely  mathematical  solution  of  the  problem,  as  he  had 
wished  to  work  out  a  practical  method  of  calculation,  but  the 
great  amount  of  time  and  labor  spent  on  this  undertaking  caused 
a  considerable  delay  of  his  publication. 

Zimmermann's  extensive  paper  is  naturally  of  a  highly  mathe- 
matical character,  and  his  penetration  into  the  subject  is  deep. 
Therefore,  we  will  give  only  the  results  of  his  investigations,  which 
may  be  applied  in  practice. 

As  has  been  said  before,  a  weightless  beam  of  uniform  sectional 


Fig.  58. 

area  and  resting  on  two  rigid  supports  is  assumed,  over  which 
moves  a  mass  with  any  constant  velocity.  If  the  mass  moves  over 
the  beam  so  slowly  that  the  pressure  against  the  beam  is  invariable, 
we  have  the  familiar  equation, 


which  expresses  the  form  of  the  deflection  curve,  Fig.  58. 

The  object  is  now  to  find  for  greater  velocities  the  path  described 
by  the  moving  mass,  in  which  case  the  pressure  against  the  beam 
is  variable.  In  other  words,  if  the  curve  shall  be  found,  it  is 
necessary  to  consider  the  effect  of  the  centrifugal  force  of  the  mov- 
ing mass.  The  differential  equation  expressing  the  desired  curve 
is 


IMPACT  ,89 

and  the  meaning  of  the  symbols  is  as  follows  :    The  time  /0  required 
by  the  mass  to  cover  one  half  of  the  span  length  /  with  the  velocity  c 

is  t0  =  -  ;  and  if  we  suppose  the  mass  were  to  descend  from  the 

G 

height  h  inside  the  same  time  /„,  then  2  h  =  gt02  =  g  —,  g  being  the 

acceleration  of  gravity. 
The  ratio 


2h  C*  P 

-=W_  <=«,and-«£ 

6  El 

E  and  /  are  modulus  of  elasticity  and  moment  of  inertia  of  the 
beam,  and  P  equals  the  weight  W  of  the  mass  m  +  the  load  Wl 
transferred  to  the  wheel  by  the  spring. 

x 

Zimmermann  puts  further  —  =  c,  and  takes  the  fall  2  h  as  a  unit 

/ 

of  measure  for  y  by  writing  rj  =  -2—  . 

2  h 

a  and  /?  are  constants,  if  P  and  c  are  assumed  to  be  invariable, 
but  Zimmermann's  method  of  integration  can  also  be  employed 
in  case  ft  is  variable,  representing  an  arbitrary  function  of  £. 

The  form  of  the  path  described  by  the  moving  mass  is  dependent 
on  a  in  a  high  degree.  This  curve  is  known  for  static  loads  where 
a  =  oo  for  c  =  o.  The  total  pressure  P  does  not  influence  a. 
If  P  changes,  then  the  ordinates  of  deflection  for  both  the  moving 
mass  and  the  static  load  change  also,  but  in  the  same  proportion. 
The  stresses  of  the  beam  are  naturally  greatest  for  the  maximal 
values  of  W  and  P. 

In  order  to  find  a  value  for  a,  which  can  be  used  in  practice, 
we  determine  first  two  limiting  values.  If  there  are  no  springs 
assumed,  we  put  W  =  P,  and 


«•     W' 

6  El 


90  SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

PI 

The  moment  Mc  at  the  center  of  the  beam  is  Mc  =  —  ;  and  if 

2 

s  is  the  stress  at  the  extreme  fiber,  and  D  the  depth  of  the  beam, 
we  have 


D 
and 


2C2S 

If  we  put  W    =  j  P  for  a  perfect  spring,  then 

6gDE 
c~s 

As  the  wheel  load  is  neither  rigidly  supported  nor  a  spring  per- 
fect, we  will  assume 


a    --       -^—  . 

C2S 

Zimmermann  has  conclusively  shown  that  for  velocities  up  to  62 
miles  per  hour,  or  91  feet  per  second,  the  path  of  the  moving  mass 
can  be  considered  as  symmetrical  about  the  center  line  of  the 
beam,  so  that  the  stress  of  the  beam  can  be  determined  from  the 
deflection  at  the  center  by  the  sufficiently  accurate  equation 


a  —  12 


Consequently  the  greatest  proportionate  increase  of  the  deflection, 
or  of  the  bending  moment,  or  of  the  stress,  is  expressed  by 

T 

Increase  = 


i ;  «  -  3 


In  the  equation  for  «,  the  quantities  g,  D,  and  c  are  expressed  in 
feet,  and  E  and  s  in  tons  per  square  inch,  a  is  independent  of  the 
length  of  the  span,  nevertheless  the  equations  can  only  be  used  for 


IMPACT  91 

very  short  spans  on  account  of  the  assumption  that  only  one  load 
moves  over  the  beam.     Assuming,  for  example, 

D  =  i  foot, 

c  =  91  feet  per  second, 

s  =8  tons  per  square  inch, 

E  =  14,500  tons  per  square  inch, 

we  find 

_  3,X32.i7X  i  X  14590  _ 
9i2X8 

which  gives  about  44  per  cent  impact.     For 

D  =  1.5  foot, 

c    =91  feet  per  second, 

5     =4  tons  per  square  inch, 

E  ==  14,500  tons  per  square  inch, 

3  X  32.17  X  1.5  X  14500  _  6 

VL    —  —    U  <  . 

9i2X  4 

which  reduces  the  impact  to  about  8  per  cent. 

A  further  result  of  Zimmermann's  investigation  can  be  summed 
up  in  the  statement  that  the  end  of  the  girder  and  its  support  in  the 
direction  of  the  motion  of  the  load  is  subjected  to  a  particular 
impact,  which  increases  in  intensity  with  the  rigidity  of  the  support, 
the  rails,  and  the  tires. 


CHAPTER   X. 
EXAMPLES    AND    CONCLUDING    REMARKS. 

A  FEW  years  ago  Professor  E.  Patton  in  Kiew,  Russia,  published 
a  book  on  secondary  stresses  in  the  Russian  language,  whose 
title,  translated  into  English  is,  "Calculation  of  Trusses  with 
Stiff  Joints,"  Moscow,  1901.  This  book  contains  a  number  of 
examples  of  bridge  trusses,  calculated  by  German  authors,  which 
Professor  Patton  collected  from  various  books  and  periodicals, 
besides  enriching  this  collection  by  a  number  of  examples  of  his 
own.  An  abstract  of  this  book  is  published  in  "Zeitschrift  fur 
Architektur  und  Ingenieurwesen,"  Hannover,  1902.  Through 
the  courtesy  of  Professor  Patton  the  writer  is  enabled  to  republish 
some  examples  of  the  above  collection,  adding  three  trusses  of 
American  design. 

•It  is  hardly  necessary  to  remark  that  although  a  truss  is  of 
foreign  origin  it  cannot  fail  to  be  serviceable  in  the  study  of 
secondary  stresses. 

The  calculation  of  secondary  stresses  is  naturally  always  pre- 
ceded by  that  of  primary  stresses,  which  is  executed  under  the 
assumption  that  the  bars  can  turn  around  frictionless  pins  and  in 
this  case  only  do  the  lines  of  stresses  coincide  with  the  axes  of 
the  bars.  But  this  assumption  is  never  fulfilled  in  our  trusses. 
Either  we  have  a  riveted  joint  or  a  pin  joint  with  frictional  resist- 
ances of  more  or  less  severity. 

As  we  have  seen  in  previous  discussions  it  is  owing  to  the  nature 
of  joints  that  the  bars  become  deformed  under  the  action  of  loads 
upon  the  truss  and  the  lines  of  stresses  displaced,  and  moreover, 
the  primary  stresses  in  a  riveted  truss  for  a  given  load  are  not 
identical  with  those  found  under  the  assumption  of  frictionless 
pins.  But  the  differences  in  the  primary  stresses  between  a 

92 


EXAMPLES    AND    CONCLUDING    REMARKS  93 

riveted  and  a  pin-connected  truss,  all  other  conditions  being 
equal,  is  inconsiderable  and  consequently  can  be  neglected  for  all 
practical  purposes. 

We  have  also  explained  that  the  calculation  of  secondary 
stresses  does  riot  need  to  take  into  consideration  the  deformations 
of  the  bars,  provided  sufficient  provision  against  buckling  has 
been  made.  In  this  case  the  stresses  in  any  given  section  of  a 
bar  are  dependent  only  on  the  bending  moment  with  respect  to 
that  section,  and  these  bending  moments  generally  reach  their 
greatest  values  at  the  ends  of  the  bars.  After  the  bending  moments 
have  been  found  we  can  then  calculate  two  different  stresses  in 
the  extreme  fibers  at  each  end  of  each  bar  according  to  known 
formulas. 

Intersection  points  of  diagonals,  which  are  riveted  at  these 
points,  are  treated  in  exactly  the  same  manner  as  any  other  panel 
points. 

The  designation  of  the  letters  in  the  tables  is  as  follows: 
/  is  the  moment  of  inertia  of  a  bar, 
/  is  the  length  of  a  bar, 
b  is  the  width  of  a  bar, 

e  is  the  distance  from  the  neutral  axis  of  a  bar  to  the  extreme 
fiber. 

The  primary  stresses  are  given  per  unit  of  area,  and  the 
secondary  stresses,  which  are  tension  on  one  side  of  the  bars 
and  compression  on  the  other  side,  are  expressed  in  percentages 
of  the  primary  stresses.  The  maximum  total  stresses  are  of 
course  obtained  by  adding  primary  and  secondary  stresses  of 
the  same  signs. 

A  study  of  secondary  stresses  proves  to  be  instructive,  for  it 
discloses  the  weaknesses  in  our  designs,  but  at  the  same  time  we 
are  also  taught  how  to  minimize  the  defects  resulting  in  stronger 
and  therefore  safer  bridges. 

Besides  the  character  of  the  truss,  which  plays  an  important 
role  in  regard  to  the  magnitude  of  secondary  stresses,  we  must 
also  point  out  the  marked  influence  on  these  stresses  exercised 


94 


SECONDARY  STRESSES   IN  BRIDGE    TRUSSES 


TABLE  2  AND  FIG.  45. 

Span    15  m.     Wrought  Iron. 


Gross 

I 

Most 

Stresses. 

1 

Member. 

area, 
sq. 

gross 

1 
cm. 

b=2C 

cm. 

danger- 
ous 

cm. 

Kg. 

Sections  in  mm. 

cm- 

cm.4 

fiber. 

sq.  cm 

% 

e 

1 

2-4 

60 

1000 

500 

19 

Top 

9-5 

-567 

14 

53 

o                             f 

U 

0, 

4-6 

60 

IOOO 

500 

19 

Top 

9-5 

-467 

18 

53 

i  nn 

H 

90  z  90  z  8 

U 

1-3 
5-7 

5° 
5° 

75° 
75° 

500 
500 

17 
17 

Bottom 
Bottom 

8-5 
8-5 

+  360 
+  300 

28 

21 

59 
59 

LjH 

£ 

80  z  80  z  9 

0 

PQ 

3-5 

70 

1125 

500 

17 

Bottom 

8-5 

-457 

18 

59 

1  =]M 

! 

i    ji 

80  x  80  z  12 

l 
b 

JL 

1-2 

5° 

79° 

350 

17    Top 

8-5 

-520 

6 

21 

nn 

6-7 

5° 

790 

35° 

17    Top 

8-5 

—  420 

4 

21 

|    HMO  | 

H-  170-  -w 

80  z  80  x  9 

1 

3-4 

30 

440 

35° 

17 

8-5 

+  100 

65 

21 

Mb 

PS 

5 

4-5 

3° 

440 

35° 

17 

8-5 

—  200 

34 

21 

r--170—  »} 

80  i  40  x.? 

2-3 
5-6 

40 
40 

610 
610 

350 
35° 

16 
16 

... 

8 
8 

+  575 
+  45° 

12 
19 

22 
22 

JL 

HQ 

I    -»]  [*10  i 

Tfi  i  75  z  8 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 


SECONDARY  STRESSES  IN  BRIDGE  TRUSSES  95 


a    O 

f£ 

'<5     ^ 

VI       P 


fr  ^ 

£  §. 

3  S 

J3      t/i 

li 


(/i  3 

PL'  C. 

§'  P 

g  g- 

O  C3 


Q 

^ 

f 


H 


-2.6m 


1 


96 


SECONDARY  STRESSES  IN   BRIDGE  TRUSSES 


TABLE  3  AND  FIG.  46. 

Span   20  m.     Wrought  Iron. 


, 

Gross 

I 

Most 

Stresses. 

Member. 

area, 
sq. 

gross 

4 

cm. 

b=  26 

cm. 

danger- 
ous 

cm. 

Sections. 

Kg. 

cm. 

fiber. 

sq.  cm. 

% 

1 

65 

4000 

400 

24 

Top 

12 

-49° 

22 

33 

<J 

Symmetrical 

§- 

3-5 

65 

4000 

400 

24 

Top 

12 

-740 

12 

33 

H 

1 

0-2 

44 

400 

400 

13 

Bottom 

6.5 

+  360 

25 

62 

u 

1 

2-4 

60 

800 

400 

16 

Bottom 

8 

+  670 

12 

50 

Symmetrical 

1 

4-4' 

60 

800 

400 

16 

Bottom 

8 

+  800 

7 

5° 

1 

b 

o-i 

60 

900 

360 

i7 

3-5 

-480 

10 

21 

•1 

2-3 

44 

400 

360 

13 

6-5 

-320 

21 

28 

Symmetrical 

1 

4-5 

44 

900 

360 

13 

6-5 

0 

... 

28 

3 

1-2 

40 

300 

360 

10 

5 

+  73° 

5 

36 

Symmetrical 

3-4 

44 

400 

360 

13 

6-5 

+  320      9 

28 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 


SECONDARY  STRESSES  IN  BRIDGE  TRUSSES 


! 3.0m. r 


98 


SECONDARY    STRESSES     IN    BRIDGE    TRUSSES 


TABLE  4  AND   FIG.  47. 

Span  36  m.     Wrought  Iron. 


Gross 

I 

Stresses. 

Most 

Member. 

area, 
sq. 

gross 
cm.-1 

1 
cm. 

b=2C 

cm. 

danger- 
ous fiber  . 

cm. 

e 

Sections. 

Kg. 

% 

cm. 

sq.  cm. 

o—  2 

86 

2040 

400 

27.2 

Top 

13.6 

-410 

66 

29 

1 

2-4 

1  66 

7110 

400 

36 

Top 

18 

-56o 

34 

22 

U 

4-6 

226 

13000 

400 

40 

Top 

20 

-600 

18 

2O 

o. 

H 

6-8 

253 

13000 

400 

40 

Top 

20 

—  630 

22 

20 

8-8' 

255 

15100 

400 

40 

Top 

20 

-660 

18 

20 

Symmetrical  and 

star  shaped, 

made  from  flats 

A-! 

48 

2140 

200 

27.6 

Bottom 

13.8 

o 

14 

and  angles. 

1 

i-3 

95 

4270 

400 

27.6 

Bottom 

13-8 

+  700 

23 

29 

u 

g 

3-5 

182 

5960 

400 

32 

Bottom 

16 

+  640 

24 

25 

o 

1 

5-7 

224 

12260 

400 

40 

Bottom 

20 

+  670 

i9 

2O 

7-9 

239 

13000 

400 

40 

Bottom 

20 

+  690 

15 

20 

b 

o-i 

96 

12800 

429 

40 

20 

+  75° 

9 

ii 

Symmetrical, 

2-3 

77 

655° 

429 

32 

.  .  . 

16 

+  700 

16 

13 

made  from 

flats. 

4-5 

56 

3660 

429 

28 

14 

+  650 

20 

IS 

cfl 

6-7 

40 

740 

429 

19.2 

9.6 

+  460 

28 

22 

Symmetrical, 

"rt 

made  from 

c 

8-9 

48 

1130 

429 

22.  2 

ii  .  i 

+  14 

429 

19 

angles. 

Q 

1-2 

136 

6920 

429 

36 

18 

-520 

35 

12 

3-4 

117 

5690 

429 

35-6 

17.8 

-45° 

38 

12 

Symmetrical  and 
star  shaped, 

5-6 

89 

3200 

429 

29.6 

14.8 

-39° 

43 

14 

made  from  flats 
and  angles. 

7-8 

55 

1640 

429 

25.2 

12.6 

-310 

32 

17 

"o5 
t3   u 

w| 

A-o 

143 

4410 

380 

25.2 

... 

12.6 

-500 

44 

15 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


99 


loo         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


TABLE  5  AND  FIG.  48. 

Span   42  m.     Wrought  Iron. 


Gross 

Most 

Stresses. 

QfAO 

I 

1 

Sections  in 

Member. 

Ctl  Ca.t, 

sq. 

gross 

cm. 

cm. 

ous 

cm. 

Kg. 

e 

cm. 

cm.4 

• 

/o 

cm. 

fiber. 

sq.  cm. 

0  -I 

1  06 

35° 

41 

Bottom 

9.0 

+  280 

5439 

r  —  r 

o'-i' 

1  06 

35° 

41 

Bottom 

9.0 

-f  160 

J939 

£           1  Web  —  40  x  1 

i  -3 

1  06 

35° 

41 

Top 

31-8 

+  280 

143  ii 

V              _l           _j|_           1  C.  PI.—  40  r  0.8 
—  flL.^          4       a  L!—  9x9x1 

"2 

i  '-3' 

1  06 

35° 

41 

Bottom 

9.0 

+  160 

22 

39 

J 

3-5 

138 

35° 

42 

Top 

34-0 

+  480 

83 

10.3 



u 

3'-5' 

138 

35° 

42 

Bottom 

7-6 

+  35° 

II 

46 

o           1  Web  -40x1 
j|           2  C.  PI.—  40x0.8 

o 

5-6 

138 

35° 

42 

Top 

34-0 

+  480 

87 

10.3 

_jb--£-         2I>9'911 

o 

5'-6' 

138 

35° 

42 

Bottom 

7-6 

+  35° 

II 

46 

> 

PQ 

6-8 

178 

350 

43 

Top 

35-8 

+  450 

III 

9.8 

}~$          1  Web  —  40  x  1 

6'-8' 

178 

35° 

43 

Bottom 

6.8 

+  400 

10 

3           2  C.  PI.—  40  x  0.8 
«f  *        1C.  PI.—  40x1 

S-v 

178 

35° 

43 

Top 

35-8 

+  45° 

45 

9.8 

-ib—  *           2^-9x9*1 

8'-9' 

178 

35° 

43   - 

Top 

35-8 

+  400 

13 

9.8 

2-4 

106 

14600 

700 

4i 

Bottom 

3i-8 

-480 

21 

22 

)                  ~              K~         I  Web  —  40  x  1 
\                                  °^           1  C.  PI.  40  x  0.8 

2>-4' 

106 

14600 

700 

4i 

Top 

9.0 

-300 

13 

78 

)                                  *?           -i  l_f  —  9x9x1 

1 

4-7 

162 

18100 

700 

42.2 

Top 

6-9 

-480 

8 

101 

)                    -  n__-_        1  Web  —  40  x  1 
(                                                1  C.  PI.-  40x1.4 

U 
a, 

4'-7' 

162 

18100 

700 

42.2 

Top 

6-9 

-35° 

9 

101 

(                                   g          1  C.  PI.—  40xO.» 
^          2  Ij-  9  x  9  x  1 

H 

7-7' 

182 

18450 

700 

43-2 

Top 

7-1 

-400 

10 

99 

*^^^*                1  Web  —  40  x  1 

—  r      ic.  PI.—  40  x  1.4 

-T          1  C.  PI.  -40x0.8 

b  == 

BI  = 

1 

g          1C.  Pl.-lOxl 

2  e 

e2 

b 

Jf         2(_!-9x9xl 

2  -3 
2'-3' 

93-6 
93-6 

712 
712 

35 
35 

17-5 

+  460 
+  320 

ii 
17 

20 

20 

|_                    1  Web  —  35  x  1 
••J^        e*         2  C.  PI—  17.  5  x  1 
\  \_        2  [f-  7  x  7  xO.9 

4-6 
4'-6' 

58.8 
58.8 

712 
712 

26 
26 

13.0 
13.0 

+  340 
+  400 

18 
15 

27 

27 

ni  Web  -  2G  x  1 
2^-8x8x1.1 

M 

7-9 

38.8 

712 

J7 

8-5 

—  260 

27 

42 

fT_  ,1         2^-8x8*1.1 

1 

7'^ 

38.8 

712 

i7 

8-5 

+  53° 

ii 

42 

"3  1 

8P 

Q 

0  -2 

114.8 

712 

40 

20.0 

-520 

44 

18 

)                                        '       1  Web  —  40  x  1 
L.  J,        2C.  PI—  20-xl 

0'~2' 

114.8 

712 

40 

2O.  O 

-300 

30 

18 

)                                      '        2|J-«x8xl.l 

3-4 

87.6 

712 

32 

16.0 

-380 

26 

22 

Jl  Web  —  32  x  1 

3  '-4' 

87.6 

712 

32 

16.0 

-310 

39 

22 

2  (J—  7  x  7  x  0.9 

6-7 

52-1 

712 

21 

10.5 

-150 

47 

34 

—  T  —       Si       21"—  10x10x1.4 

6'-7' 

52-  I 

712 

21 

10.5 

—  420 

T4 

34 

T.—  A     L 

I  -2 

21  .  I 

600 

15 

8.0 

+  590 

12 

40 

JM 

4-5 

21  .  I 

600 

J5 

8.0 

+  420 

17 

40 

0 

'£ 

7-8 

7'-8' 

21  .  I 
21  .  I 

600 
600 

15 
12 

8.0 
8.0 

+  575 

+  100 

5 
90 

40 
40 

I-"~|              2  Ij-  7  x  7  x  0.3 

4'-5' 

21  .  I 

600 

IS 

8.0 

+  100 

60 

40 

I  '-2' 

21  .  I 

600 

15 

8.0 

+  100 

65 

40 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES         IOI 


IO2 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


Verticals 

Diagonals 

Bottom  Chord 

On         Oo           M 

^        I          ± 

4x                                                    M 

4^                                         K)                                  O 

ON                                      4^-                                  to 

to        to        to 

to         to         to 

00          00          00 

Oo                                 4*. 
0                                -J 

to                                   Oo 

4*                                    ON 

0?                                   ^J                              ^ 

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SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


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SECONDARY    STRESSES    IN    BRIDGE    TRUSSES          105 

by  the  dimensions  of  each  bar  and  the  distribution  of  its  material. 
It  is  for  this  reason  that  each  table  contains  a  column,  which 
gives  the  ratio  between  length  and  width  of  each  bar  having  a 
symmetrical  section,  and  for  bars  with  unsymmetrical  sections 
the  ratio  between  length  and  distance  of  outer  fiber  from  the  neu- 
tral axis  is  given,  and  with  a  few  exceptions  the  tables  show  also 
the  distribution  of  material  in  the  sketches  of  the  sectional  areas. 

We  will  now  point  out  some  facts  which  indicate  how  different 
trusses  are  affected  by  secondary  stresses.  An  inspection  of  the 
Tables  II,  III  and  IV,  with  Figs.  45,  46  and  47,  shows  that  the 
secondary  stresses  of  the  chords  in  single  Warren  trusses  without 
verticals  increase  from  the  center  of  the  span  toward  the  end,  while 
for  the  web  members  these  stresses  increase  in  the  opposite  direc- 
tion. This  circumstance  is  in  so  far  fortunate  as  the  web  mem- 
bers in  the  neighborhood  of  the  center  of  the  span  have  often  a 
surplus  in  section  and  the  end  members  of  both  the  top  and 
bottom  chords  show  this  surplus  still  oftener. 

The  secondary  stresses  in  Warren  trusses  with  verticals,  Fig. 
48,  Table  V,  and  Fig.  49,  Table  VI,  do  not  entirely  bear  out 
the  statements  made  in  reference  to  the  Warren  trusses  without 
verticals.  It  is  the  bottom  chord  which  shows  in  both  cases 
irregularities  not  observed  in  Warren  trusses  without  verticals. 
The  run  of  the  secondary  stresses  in  Fig.  48  and  Table  V,  can 
easily  be  accounted  for  by  the  fact  that  the  live  load  covers  only 
one-half  of  the  span. 

If  verticals  carry  a  heavy  concentrated  load,  then  they  elongate 
a  good  deal.  These  elongations  are  the  cause  of  considerable 
deformations  in  the  bottom  chord  members  accompanied  by 
severe  secondary  stresses. 

The  bottom  chord  member  4-6  in  Fig.  49,  shows  the  greatest 
secondary  stress  to  be  43  per  cent  of  the  primary  stress,  while  in 
the  members  1-3  of  Fig.  48,  it  is  as  high  as  143  per  cent.  In  com- 
paring these  secondary  stresses  with  respect  to  the  unit  primary 
and  total  stresses,  we  find  that  the  secondary  stress  in  the  member 
1-3  of  Fig.  48  amounts  to  5690  pounds  per  square  inch,  which 


106  EXAMPLES    AND    CONCLUDING    REMARKS 

with  the  primary  stress  of  3980  pounds  per  square  inch,  results  in 
9670  pounds  per  square  inch,  while  in  the  member  4-6  of  Fig. 
49,  the  secondary  stress  is  5160  pounds  per  square  inch,  making  a 
total  of  12,000  +  5160  =  17,160  pounds  per  square  inch.  But  the 
bottom  chord  section  6-8  of  Fig.  48,  with  an  increase  of  stress  of 
in  per  cent  or  7100  pounds  per  square  inch  shows  a  primary 
stress  of  6400  pounds  per  square  inch,  which  gives  a  total  of 
13,500  pounds  per  square  inch  or  a  stress  considerably  higher  than 
in  the  member  1-3  of  the  same  truss. 

Of  course  the  idea  presents  itself  as  natural  that  the  reduction 
of  these  high  secondary  stresses  is  easily  affected  by  providing 
the  verticals  with  ample  sectional  areas,  because  in  so  doing  we 
reduce  their  elongations  and  consequently  the  deformations  of 
the  bottom  chord. 

Excluding  the  truss  Fig.  51  and  Table  VIII  from  a  comparison 
on  account  of  the  incompleteness  of  the  data,  we  find  the  run  of 
the  secondary  stresses  in  the  two  Pratt  trusses  Fig.  50,  Table  VII, 
and  Fig.  52,  Table  IX,  in  a  rather  close  agreement,  although 
these  stresses  have  been  determined  under  varying  conditions  and 
such  differences  as  do  exist  can  be  explained. 

The  secondary  stresses  for  the  truss  Fig.  50  and  Table  VII, 
have  been  calculated  for  one  single  position  of  the  live  load,  while 
those  for  the  truss  Fig.  52  and  Table  IX,  have  been  found  by  the 
method  of  influence  lines.  Sometimes  the  writer  reversed  the 
direction  of  the  train,  but  he  always  placed  the  second  driver  of 
the  first  engine  at  any  one  of  the  panel  points,  so  that  the  stresses 
given  are  not  the  absolute  greatest. 

The  secondary  stresses  from  the  center  of  the  span  along  the 
top  chord  and  the  end  post  in  Fig.  50  first  increase  and  then 
decrease,  and  this  is  also  true  for  the  truss,  Fig.  52,  up  to  the  inter- 
section point  between  end  post  and  collision  strut,  the  maximum 
stress  being  reached  in  the  lower  fragment  of  the  end  post.  This 
maximum  is  due  to  the  collision  strut,  which  by  its  division  of  the 

end  post  into  two  fragments  decreases  the  ratio  -  and  has,  there- 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


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SECONDARY    STRESSES    IN    BRIDGE    TRUSSES         115 


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Il6        SECONDARY    STRESSES    IN    BRIDGE   TRUSSES 


TABLE  10  AND  PIG.  53. 

Span  27  m.     Wrought  Iron. 


Gross 

3TC3. 

I 

1 

b 

Most 

Stresses. 

1 

Case  I. 

Case  H. 

Member. 

sq. 

gross 
cm4. 

cm. 

cm. 

danger- 

ous  fiber. 

cm. 

Kg. 

Max. 

e 

Sections. 

cm. 

sq.  cm. 

% 

Kg. 

% 

sq.  cm. 

i-3 

60 

2800 

45° 

23 

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83 

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56 

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ex 

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4500 

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25 

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21  .  I 

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177 

-30.0 

88 

21 

T 

Sections  con- 

"2 

0-2 

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2800 

45° 

23 

Top 

15-8 

+  18.8 

41 

+  18.8 

41 

28 

sist  of  i  web, 

0 

2  angles  and 

U 

i  or  2  or  no 

g 

2-4 

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3700 

45° 

24 

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l8.7 

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24 

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« 

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25 

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21  .  I 

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127 

+  33-8 

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21 

b  =  2e 



b 

1-2 

36 

2100 

75° 

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13 

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12 

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10 

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23 

37 

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400 

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7 

0 

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54 

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9 

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33 

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25 

42 

II 

o-i 

96 

1400 

600 

16 

8 

0 

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24 

37 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES         117 


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Il8         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


TABLE  11  AND  FIG  54. 

Span   27  m.      Wrought  Iron. 


Stresses. 

Gross 
area, 

I 

1 

b 

Most 

e 

1 

Case  I. 

Case  II. 

Member. 

sq. 
cm 

gross 
cm4- 

cm 

cm 

danger- 
ous fiber. 

cm. 

Kg. 

Max. 
Kg 

e 

Sections. 

i 

/O 

/O 

sq.  cm. 

sq.  cm. 

"g 

1-3 

60 

2800 

45° 

23 

Top 

7-2 

0 

-12.5 

27 

62 

0 

U 

OH 

3-5 
5-7 

80 

IOO 

3700 
4500 

45° 
45° 

24 

25 

Bottom 
Bottom 

18.7 

21  .  I 

-18.8 
-15.0 

55 
227 

-28.1 
—  30.0 

36 

24 

21 

T 

"g 

0-2 

60 

2800 

45° 

23 

Top 

15-8 

+  18.8 

73 

+  18.8 

73 

29 

Sections   con- 
sist of  i  web, 

I 

2  angles  and 

U 

i  or  2  or  no 

o 

2-4 

80 

3700 

45° 

24 

Top 

l8.7 

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IO2 

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IOO 

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115 

+  33-8 

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21 

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b 

1—2 

36 

2100 

375 

26 

13 

0 

... 

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28 

14 

3-4 

28 

900 

375 

20 

10 

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56 

+  22.3 

56 

19 

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5-6 

43 

400 

375 

14 

7 

0 

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9° 

27 

tj 
•fi 

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0-3 

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29 

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10 

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21 

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96 

1400 

600 

16 

8 

0 

—  10.4 

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37 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


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120         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


TABLE  12  AND  FIG    55. 

Span  40  m.     Wrought  Iron. 


Case  I.            Case  II. 

Sections  in  cm, 

Stresses, 

1 

b 

Most 

e 

when  diagonals  are 

1 

Member. 

cm. 

cm. 

danger- 
ous fiber 

cm. 

riveted. 

not  riveted. 

e 

Kg. 

Kg. 

sq.  cm. 

/o 

sq.  cm. 

/o 

--3 

400 

46.2 

Bottom 

36.5 

—  1501    117 

-So 

120 

ii 

i 

i 

3-5 

400 

46.2 

Top 

9-7 

~39° 

12 

-39c 

12 

41 

i 

1  Web 

—  45  x  1.5 

1  C.  P 

.—  48  x  1.2 

2LS— 

11.8  x  11.8  E  1.3 

1 

5-7 

400 

47-4 

Top 

8.6 

-460 

12 

-460 

12 

46 

r"I 

U 

a, 

8 

o 

h 

-_ 

>_ 

1  We 

).—  45  x  1.5 

2  C.I 

M  —  48  x  1.2 

2  L8—  11.8  x  11.8  s.1.3 

7-9 

400 

48.6 

Top 

8.0 

-500 

7 

-500 

7 

5° 

"| 

1 

9-1  1 

400 

48.6 

Top 

8.0 

-540 

8 

-540 

8 

5° 

1 

L 

1  Web 

-  45  x  1.5 

3  C.  P 

.  -  48  x  1.2 

se- 

11.8 x  11.8  x  l.a 

12 

0-2 

400 

40.2 

Top 

36.5      +210 

145 

+  210 

117 

1  1 

same  as  lor  1—3 

1 

2-4 

400 

46.2 

Bottom 

9-7 

+  480 

7 

+  480 

8 

4i 

and  4-5 

u 

| 

4-6 

400 

47-4 

Bottom 

8.6 

+  530 

8 

+  53° 

10 

46 

same  as 

for  5-7 

§ 

pq 

6-8 
8-10 

400 
400 

48.6 
48.6 

Bottom 
Bottom 

8.0 
8.0 

+  560 
+  590 

7 
7 

+  560 

6 

6 

5° 

.same  as  for  7-9 
and  o-n 

u  

18  --•* 

w 

o-i 

400 

48 

-105 

133 

-105 

124 

l 
b 

8 

t*1 

-5 

.43  x  1.5 

.y 

a  PU- 

48 x  1.3 

"Tj 

SH.— 

11  x  1 

I> 

4L8_10.5  x  10.5  x  1.8 

2-3 
4-5 
6-7 

400 
400 
400 

i7 

.... 

-3° 

"1° 

—  60 

483 
220 
141 

-bo 

220 
141 

24 
24 
24 

fi 

8-9 

400 

17 

-70 

50    -70 

24 

i 

10-11 

400 

17 

-90 

o    —90 

0 

24 

4  ,?_R  x  8  x  1 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 

SECONDARY    STRESSES    IN    BRIDGE    TRUSSES         121 


TABLE  13  AND  FIG.  55. 

Span   40  m.      Wrought  Iron. 


Case  I.                Case  II. 

!           i 

Stresses, 

Gross  ; 

when  diagonals  are 

Member. 

area,  |     1 
sq.       cm. 

b     iet  =  e2 
cm.      cm. 

riveted. 

not  riveted. 

Sections  in  cm. 

cm. 

Kg. 

07 

1 

Kg-    L 

l 

sq.  cm. 

/O 

b 

"  "  i  /C 
sq.  cm. 

b 

3     ''-2 

84 

56630        15 

+  543    23  '9.  4 

+  543 

II 

.0 

L—  so—  *1      SF1'30«1-4 

bJD 

c 

-5 

C  *j 

3-4 

73 

566 

28          14 

+  500 

22 

10 

+  5°° 

20 

2C 

j^__28__      i        2  Fl.  28  x  1.3 

UO  .IJP 

C«    J^ 

5-6 

60 

566 

25 

I2-5 

+  420 

23 

ii 

+  420 

17 

2  3 

_  -2ru25xl.a 

IT 

7-8 

60 

566 

25 

12-5 

+  235 

34 

1  1 

+  235 

25 

23 

U--  25-J 

5 

9-10 

49 

566 

18.2 

9.1 

+    90 

78 

16 

+  90 

33 

3] 

nr  1^^ 

o-3 

95 

566 

21 

10.5 

-580 

16 

13 

-580 

9 

27 

!            1  I           ]          4  Ls_10.5  at  10.5  xf.2 

nr 

W---19--*) 

2 

2-5 

85 

566 

19 

9-5 

-500 

14 

15 

—  500 

14 

So 

| 
||         !           4  L8—  9.5  x.9.5  it  1.2 

o 

nr 

'C 

c 

§ 

4-7 

72 

566 

19 

9-5 

-440 

16 

15 

-440,  1  1 

,y< 

*---  19--*i 

•S 

c/j 

6^ 

72 

566 

I9 

9-5 

-275!   24    15 

-275 

18 

3° 

4Cf-9.Sx9.5il 

1 

o 

_    i 

nr 

% 

i 

5 

' 

w—  T8.2-»l 

8-1  1 

49 

566 

18.2 

9-i 

-155 

40 

16 

-^ 

16 

S' 

1  Fl.  0.5  x'1.2 

2  L3-8.&  i  8.5  x  1.2 

122         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


a 


o 


EXAMPLES    AND    CONCLUDING    REMARKS  123 

fore,  no  beneficial  influence  on  the  end  post  as  far  as  secondary 
stresses  are  concerned. 

The  run  of  the  secondary  stresses  in  the  web  members  for  both 
trusses  is  very  nearly  the  same,  increasing  from  the  end  toward 
the  center  of  the  span. 

The  vertical  5-6  in  Fig.  52  has  been  assumed  as  carrying  no 
dead  weight,  otherwise  the  percentage  of  secondary  stress  would 
be  very  high. 

Comparing  the  secondary  stresses  in  the  bottom  chords  of  Fig. 
50  and  Fig.  52,  we  see  that  they  decrease  in  Fig.  50  from  the  end 
toward  the  center  of  the  span,  while  they  increase  in  Fig.  52  in 
the  same  direction.  This  phenomenon  is  attributable  to  the 
collision  strut,  which  is  wanting  in  Fig.  50.  This  strut  resists  the 
hip  vertical  in  its  elongation  and  consequently  lessens  the  defor- 
mations of  the  bottom  chord  sections  0-2  and  0-4. 

The  middle  vertical  5-6  in  Fig.  52,  and  the  collision  strut  experi- 
ence both  a  reversal  of  stresses  and  this  is  quite  natural,  as  these 
members  are  supposed  to  carry  neither  dead  nor  live  load 
stresses. 

The  double  intersection  Warren  truss  without  verticals,  Fig. 
53  and  Table  X,  as  also  Fig.  54  and  Table  XI,  shows  rather  severe 
secondary  stresses,  which  increase  from  the  end  of  the  span 
towards  the  center  and  this  direction  is  contrary  to  that  shown  by 
the  single  Warren  truss.  The  reason  for  these  high  secondary 
stresses  is  that  for  unequal  loading  of  the  single  systems  which 
compose  the  truss,  the  chords  are  subjected  to  rather  large  defor- 
mations. These  latter,-  and  consequently  the  secondary  stresses, 
can  be  reduced  by  the  insertion  of  verticals,  which  effectively 
connect  the  two  single  systems.  The  truss  shown  in  Fig.  55  and 
Tables  XII  and  XIII  proves  this  clearly.  The  secondary  stresses 
in  the  chords  of  these  trusses  increase  from  the  center  of  the  span 
toward  its  end  and  those  of  the  diagonals  increase  from  the  end 
toward  the  center  of  the  span,  which  is  the  same  run  as  found  in 
single  Warren  trusses.  The  stresses  in  the  diagonals  also  increase 
when  they  are  riveted  at  their  intersection  points. 


124         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


TABLE  14  AND  FIG.  56. 

Span  27.02   m.     Steel. 


Case  I. 

Case  II. 

Stresses. 

Mem- 
ber. 

Gross 
area, 
sq. 

I 

gross 
cm.4 

1 
cm- 

b 
cm. 

Most 
danger- 
ous fiber  . 

e 
cm. 

1 
e 
cm. 

For    each 
uniform 
total    load- 

For the 
most  unfa- 
vorable po- 
sitions of 

Sections. 

cm.   | 

ing. 

the  loads. 

% 

Max. 

% 

I 

I 

Kg. 

I 

i 

net. 

gross. 

sq.  cm. 

net. 

o-i 

239 

43736 

386 

47-6 

Top 

10.6    37 

33 

28 

-591 

27 

-t       p  f 

1 

1-3 

239 

43736 

386 

47-6 

Top 

10.  6 

37 

29 

25 

-582 

tz  i 

CJ 

(X 

3-5 

239 

43736 

386 

47-6 

Top         10.6    37 

7 

6 

-559 

7 

1   Li 

0 

2  WeBte  18'  x  % 

H 

5-5' 

239 

43736 

386 

47-6 

Top 

10.6    37 

7 

6 

-557 

5 

•iC.  P1.18,"*  %*. 

1 

" 

0-2 

216 

40757 

213 

47 

Bottom 

ii 

19 

40 

33 

+  658 

42 

«—  10_» 

0 

j 

5 

2-4 

216 

40757 

410 

47 

Bottom 

ii 

37 

36 

30 

+  7i4 

38 

1 

0 

4-6 

216 

40757 

395 

47 

Bottom 

ii 

45 

8 

7 

+  651 

6 

J       L  ? 

1 

6-8 

216 

40757 

387 

47 

Bottom 

ii 

35 

8 

7 

+  632 

7 

Z  \Veba  18*  x  % 
1  C.  PI.  20J4*  i  %* 

4  L'_;3*;x  »*'.  H 

1 

e2 

b 

1-2 

55 

688 

213 

16 

Bottom 

8 

J3 

60 

47 

-332 

54 

s 

1-4 

55 

688 

298 

16 

Top 

8 

19 

76 

59 

-432 

86 

J       L 

a 

a 

-~i      r 

1 

•3-4 

55 

688 

298 

16 

Bottom 

8 

i9 

80 

62 

-34o 

52 

•      i 

4L'_3*,  3*i  % 

Q 

'3-6 

55 

688 

365 

16 

Top 

8 

23 

28 

22 

-459 

23 

'5-6 

55 

688 

365 

16 

Bottom 

8 

23 

35 

27 

-396 

18 

5-8 

55 

688 

389 

16 

Top 

8 

24 

49 

38 

-436 

27 

'     The  direct  stresses  and  the  sums  of  the  direct  and  secondary  stresses  have  been 
each  calculated  for  the  most  unfavorable  positions  of  the  live  load. 


SECONDARY    STRESSES    IN    BRIDGE    TRUSSES         125 


i? 


g.w 

"^  HI 


9- 


9-4 


H 

e-S 


9—* 

I 

i 

T 
i 


1 


126          SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 


TABLE  15   AND  FIG.  57. 

Draw   Span   202   Feet.      Single  Track.      Steel. 


Gross 

Most 

Stresses. 

1 

Sections. 

Member. 

area, 

danger- 

Ibs 

sq. 

gross. 

in. 

in. 

in. 

erf 

e 

in. 

sq.  in. 

/o 

"O   w 

o-i 

23-45 

500 

452 

12.44 

4.  12 

Top 

-6080 

g 

log 

1  C.  PL—  20"  x 

5^f 

1-3 

23-45 

500 

303 

12.44 

8.32 

Bottom 

4300 

22 

36 

~ 

l| 

—  _ 

0 

3-5 

23.45 

500 

12  .  44 

4.  12 

Top 

—  4300 

78 

72 

js> 

00 

U 

1 

1 

^  -  12  U  — 

i,, 

T 

£ 

b 

"" 

5-7 

20.58 

359 

3i8 

12 

6 

+  7210 

80 

26 

—  • 

.1 

— 

0 

O—  2 

17.64 

323 

3°3 

12            6 

+  61^0 

4 

25 

.-rfB-, 

0* 

U 

£ 

§ 

PQ 

2-4 

4-6 
6-8 

17.64 
17.64 
17.64 

323 
323 
323 

3°3 
3°3 
3°3 

12 
12 
12 

6 
6 
6 

... 

+  6120 

+  1020 
+  IO2O 

1  6 
1186 
868 

25 
25 

25 

LI 

2  —  12"l!l—  25it- 

1-4 

14.70 

288 

452 

12 

6 

+    610 

190 

38 

j 

c: 

.i 

B 

Diagonal 

4~5 

17.64 

323 

452 

12 

6 

... 

+  8120 

60 

38 
1 

II 

e 

1-3 

r 

5-8 

26.40 

55° 

452 

12.44 

4-38 

Top 

-7900 

40 

103 

!™12"lT-30ff.. 

1 

I  C.  Plr—  20"x 

Jfi 

1 

b 

1-2 

13.68 

120 

33612.38 

6.19 

+  8200 

3 

27 

(/3 

3-4 

13.68 

120 

336 

12.38 

6.19 

.  .  . 

1280$ 

27 

H 

sq.in. 

^y 

5-6 

13.68 

120 

336 

12.38 

6.19 

+  8200 

IIO 

27 

o 

2  Vi 

eb.-l9S/ 

*  ^ 

16 

> 

rr=_ 

• 

7-8 

28.54 

1475 

432 

20 

10 

-2750 

•VI 

t=T- 

• 

Secondary  stresses  are  calculated  for  gross  moments  of  inertia. 


SECONDARY    STRESSES    IN  BRIDGE    TRUSSES          127 


ct> 
2   S 


it 

g     3 

°-o 


=  a 

"• 


a 

qq 

Oi 

-j 


% 


•* 


I* 


^___Xr-36-0-c-^_-/^| 


---23-0-c-c J 


128          SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

The  parabola  truss,  Fig.  56  and  Table  XIV  is  a  Russian  railroad 
bridge  with  the  wooden  ties  resting  directly  on  the  top  chords. 
The  ratios  between  total  stress  and  primary  stress  remain  constant 
for  any  uniformly  distributed  load. 

The  single  Warren  truss  with  verticals,  Fig.  57  and  Table  XV, 
is  continuous  over  three  supports  and  intended  for  a  drawbridge. 
The  writer  determined  the  secondary  stresses  for  "  bridge  closed," 
the  truss  resting  on  three  supports  and  covered  with  the  live  load 
from  end  to  end.  The  center  reaction  has  been  taken  as  the 
unknown  quantity  and  calculated  by  means  of  the  principle  of 
the  derivative  of  work.  After  this  reaction  was  found  the  primary 
stresses  were  calculated. 

Owing  to  the  fact  that  this  truss  is  continuous  over  three  sup- 
ports, the  run  of  the  secondary  stresses  is  contrary  to  that  in  single 
Warren  trusses  on  two  supports.  We  see  that  they  increase  in 
the  chords  from  the  end  of  the  span  toward  the  center  and 
decrease  in  the  diagonals  in  the  same  direction.  High  secondary 
stresses  can  be  expected  at  the  center  of  the  span  and  in  its  neigh- 
borhood, no  matter  whether  the  bridge  is  closed  or  open,  as  it  is 
at  these  places  where  the  deformations  are  the  greatest,  and  they 
are  the  smallest  in  the  end  panel  with  inconsiderable  bending 
stresses  when  the  bridge  is  open. 

An  increase  in  the  stress  of  the  bottom  chord  section  4-6  of 
1 1 86  per  cent  is  very  severe,  but  we  must  not  forget  that  its  primary 
stress  is  very  small,  amounting  only  to  1020  pounds  per  square 
inch.  The  primary  stress  in  the  end  bottom  chord  sections  is 
six  times  greater  than  that  in  the  bars  4-6  and  6-8  and  conse- 
quently offers  a  greater  resistance  to  the  deformations  at  the 
panel  point  2.  The  total  stresses  per  square  inch  in  the  bottom 
chord  sections  from  the  end  of  the  span  to  the  center  are  6360, 
9790,  13,120  and  9870  pounds,  while  the  primary  stresses  amount 
to  6120  and  1020  pounds  per  square  inch. 

We  will  now  enumerate  some  points  which  are  guiding  for  a 
designer  to  minimize  secondary  stresses  due  to  static  loads.  The 
effects  of  impact,  vibrations,  derailments  and  collisions  are  beyond 


EXAMPLES    AND    CONCLUDING    REMARKS  129 

an  analysis,  but  they  should  not  be  overlooked  because  they  too 
are  of  a  secondary  nature. 

It  has  already  been  mentioned  that  secondary  stresses  decrease 
if  the  ratio  between  the  length  of  a  bar  and  its  width  increases. 
This  means  that  members  of  great  circumference  and  shortness 
are  more  susceptible  to  secondary  stresses  than  long  and  slender 
members,  a  fact  already  noted  when  we  spoke  of  secondary  stresses 
in  cross  frames.  In  this  respect  a  truss  with  a  curved  chord  would 
be  a  good  selection,  as,  for  instance,  the  parabolic  truss  or  the 
Schwedler  truss  where  the  web  members  are  particularly  of  light 
sections.  The  double  intersection  Warren  truss  is  unfavorable,  as 
we  have  "seen,  but  it  can  be  improved  by  the  insertion  of  verticals, 
which  connect  the  two  single  systems  in  an  effective  manner. 

If  in  multiple  intersection  trusses  the  single  trusses  act  more  or 
less  independent  of  each  other,  we  may  predict  high  secondary 
stresses,  because  in  such  cases  the  wave  like  deformations  of  the 
chords  are  rather  large. 

The  secondary  stresses  in  continuous  trusses  over  three  supports 
are  very  severe  at  the  center  and  in  its  neighborhood  and  for  this 
reason  pins  at  these  points  appear  to  be  desirable,  in  order  to 
reduce  the  stresses. 

The  width  of  a  member  in  the  plane  of  the  truss  should  not 
be  greater  than  buckling  and  good  connections  dictate.  The 
transverse  width  of  verticals  to  which  floorbeams  are  riveted, 
should  also  be  kept  inside  proper  limits,  as  otherwise  the  secondary 
stresses  caused  by  the  floorbeams  may  prove  to  be  very  high. 
The  removal  of  the  material  from  the  axis  of  the  member  to  its 
periphery  gives  the  required  moment  of  inertia  in  the  most 
economical  way. 

Suspenders  should  be  liberally  proportioned  to  avoid  great  elon- 
gations and  consequently  large  deformations  of  the  bottom  chords. 

Strict  attention  should  be  paid  to  good  detailing  of  the  panel 
points ;  the  axes  of  the  bars  must  lie  in  the  plane  of  the  truss  and 
eccentricities  should  be  avoided  as  far  as  practicable. 

Curved   members  should  not  be    tolerated  under  anv  circum- 


130         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

stances,  and  brackets  on  posts  may  be  used  only  when  nothing 
better  can  take  their  place. 

The  movable  ends  of  bridges  must  be  kept  in  proper  working 
order,  otherwise  they  will  be  the  cause  of  unnecessary  stresses. 

The  use  of  collision  struts  cannot  be  recommended  as  far  as 
secondary  stresses  are  concerned,  because  they  divide  the  end 
posts  into  two  fragments,  decreasing  the  ratio  between  length  and 
width  of  these  members  and  increasing  the  secondary  stresses. 
The  collision  struts  are,  as  a  rule,  rather  weak  members  and  it  is 
a  question  whether  it  would  not  be  better  to  use  the  metal  on  the 
end  posts  instead  of  on  these  struts,  increasing  their  strength  in 
the  direction  of  the  plane  of  the  truss  as  well  as  also  at  right  angles 
to  this  plane.  It  is  very  essential  to  connect  the  end  posts  by  a 
substantial  bracing,  designing  the  connections  in  the  best  manner 
possible,  and  moreover,  if  a  weak  end  bracing  is  used  the  money 
spent  on  the  horizontal  top  bracing  is  wasted. 

The  position  of  the  horizontal  bracings  should  be  selected  with 
a  view  of  reducing  eccentricities. 

On  account  of  the  riveted  connections  between  the  floorbeams 
and  the  main  trusses,  the  secondary  stresses  in  the  vertical  posts 
of  the  trusses  very  often  exceed  by  far  the  limit  of  stress  set  by 
the  specifications.  Generally  speaking  this  question  has  not  yet 
been  thoroughly  settled.  A  number  of  suggestions  have  been 
made  to  remedy  the  defect  and  also  partly  carried  out.  Deep 
floorbeams  tend  to  reduce  the  secondary  stresses,  but  they  cannot 
always  be  made  deep  enough  to  be  effective  from  want  of  depth 
in  the  floor,  and  besides,  if  they  are  very  deep,  they  may  exceed  the 
limit  of  economy.  The  suggestion  to  give  the  floorbeams  a  down- 
ward camber,  as  shown  in  Fig.  59,  originated  with  Professor 
Engessor.  It  is  clear  that,  if  a  floorbeam  is  given  a  deformation 
corresponding  to  the  load  it  has  to  carry  and  afterwards  riveted 
between  the  main  trusses,  it  will  be  quite  effective  in  reducing  the 
secondary  stresses  in  the  posts.  Another  way  to  avoid  these 
stresses  in  the  verticals  would  be  to  give  the  floorbeams  on  the 
main  trusses  free  supports,  provided  care  is  taken  to  properly 


11 


EXAMPLES    AND    CONCLUDING    REMARKS  131 

transmit  the  wind  and  braking  forces  to  the  main  trusses.  An 
important  example  to  accomplish  the  object  in  question  is  fur- 
nished by  the  double  track  railroad  arch  bridge  across  the  Rhine 
river  at  the  city  of  Worms,  Germany.  The  river  spans  consist 
of  two  shore  spans,  each  about  351  feet  long  and  one  central  span 
of  about  388  feet.  The  main  trusses  or  arch  ribs  produce  vertical 
reactions  only  owing  to  a  tension  member  running  along  the  floor, 
which  intersects  with  the  main  trusses.  The  floor  is  carried  by 
the  main  trusses  by  means  of  stiff  suspenders 
to  which  the  floorbeams  are  pin-connected. 
The  design  of  the  floor  is  such  that  it  is  fixed 
transversely,  not  by  riveting,  but  by  abutting 
ends,  and  longitudinally  it  is  fixed  exclusively 
at  the  center  of  the  span.  The  main  trusses 
and  all  of  the  bracings  are  riveted  work. 
The  intermediate  cross  bracings  are  attached 
to  the  main  trusses  by  means  of  plates,  in 

such   positions   that  they  offer  only  a  very  ; " 

small  resistance  to  the  deformations  of  the 
trusses  in  vertical  planes.  These  arrangements  of  the  details 
allow  a  cross-section  for  unequal  loading  of  the  bridge  to  take 
the  shape  of  a  rhomboid.  Here  we  see  then  that  the  reduction 
of  secondary  stresses  in  the  suspenders  is  aimed  at  by  the  use  of 
pins  and  a  skillful  design  of  the  bracings  and  their  attachments.* 

*  Our  practice  of  stiff  connections  between  floorbeams  and  main  trusses  seems 
to  be  well  founded,  as  loose-jointed  cross  constructions  may  prove  to  be  more 
injurious  to  a  bridge  than  severe  secondary  stresses.  But  this  does  not  mean 
that  we  should  leave  the  secondary  stresses  to  take  care  of  themselves;  on  the 
contrary,  proper  attention  should  be  paid  to  these  with  a  view  to  their  reduction. 
Deep  floorbeams,  substantial  gussets  and  a  generous  number  of  rivets  to  connect 
the  floorbeams  to  the  main  trusses  are  desirable,  as  also  posts,  whose  width  longi- 
tudinally and  transversely  is  kept  within  proper  limits. 

Although  these  points  are  well  known  to  experienced  bridge  engineers  never- 
theless the  German  government  called  attention  to  them  in  printed  circulars, 
issued  1904,  for  the  observance  of  those  engineers  who  are  charged  with  the  de- 
signs of  engineering  structures. 

It  appears  from  these  circulars  that  the  German  government  is  inclined  toward 
stiff  connections  between  floorbeams  and  main  trusses  and  that  it  discourages  the 
designs  of  hinged  floor  systems  in  so  far  as  it  requests  a  proof  of  their  advantages. 


132         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

It  is  faulty  to  design  the  top  chord  flanges  of  the  floorbeams 
exclusively  for  vertical  loads,  if  at  the  same  time  they  are  also 
charged  with  the  duty  of  transmitting  the  braking  forces  to  the 
main  trusses,  which  they  can  only  do  by  being  bent  in  a  horizontal 
plane.  In  such  cases  the  lloorbeam  flanges  should  be  provided 
with  brackets,  as  is  shown  in  Fig.  60,  or  any  other  suitable  con- 
struction may  be  designed  to  prevent  the  bending  of  these 
flanges. 

We  believe  the  reduction  of  secondary  stresses  by  the  use  of 


Main  Truss 


Main  Truss 


Fig.  60. 


pins  has  been  overestimated  and  that  in  general  the  diameters 
of  the  pins  are  too  great.  Secondary  stresses  require  the  pins  to 
be  as  small  in  diameter  as  is  consistent  with  the  strength  and 
safety  of  a  bridge.  Attention  should  be  paid  to  have  the  sur- 
faces, which  are  in  contact  with  each  other,  very  smooth  to  facili- 
tate turning. 

It  has  been  suggested  that  light  colored  paints  would  be  pref- 
erable to  dark  ones,  as  they  absorb  less  heat  and  consequently 
reduce  the  temperature  stresses. 

For  the  reduction  of  the  effects  of  impact  and  vibrations  the 
following  points  deserve  consideration.  Since  the  mass  of  a  body 
which  receives  a  blow  must  be  great  if  the  effects  of  a  blow  shall 
be  small,  it  is  therefore  beneficial  to  use  heavy  floor  construction 
and  a  heavy  track.  Full  webbed  floorbeams  and  trackstringers 
and  a  ballasted  track  diminish  the  range  of  vibrations. 

From  economical  reasons  the  ties  are  often  placed  directly  on. 


EXAMPLES    AND    CONCLUDING    REMARKS  133 

the  top  chords,  when  the  use  of  a  steel  floor  would  be  much 
better. 

Tension  members  should  be  built  stiff,  as  it  is  within  the  range 
of  possibilities  that  such  members,  for  instance  suspenders,  are 
subjected  to  compression  in  consequence  of  vibrations. 

Riveted  connections  and  stiff  members  throughout  the  bridge 
are  in  so  far  of  advantage  as  they  tend  to  decrease  the  kinetic 
energy  of  vibrations.  The  riveting  of  the  members  at  their  inter- 
section points,  the  introduction  of  secondary  and  even  redundant 
members,  all  tend  to  diminish  the  time  and  the  amplitude  of 
vibrations. 

To  further  decrease  the  effects  of  moving  loads,  it  is  necessary 
to  keep  the  track  in  perfect  working  order.  The  passage  from 
the  road  bed  to  the  bridge  should  be  smooth.  Great  attention 
must  be  paid  to  the  rail  ends  at  the  splices.  If  these  rail  ends 
are  not  of  the  same  height  they  will  be  the  cause  of  severe  shocks, 
and  particular  stress  should  be  laid  on  the  tightness  of  the  bolts 
at  the  rail  splices.  Long  rails  are  naturally  of  advantage,  since 
they  reduce  the  number  of  splices,  but  on  a  bridge  of  short  span 
there  should  be  no  rail  splice  at  all. 

In  reference  to  derailments  and  collisions  we  will  mention  a 
few  points  which  are  deserving  of  consideration  from  the  side  of 
the  designer.  Of  course,  it  cannot  be  his  object  to  design  a  bridge 
so  that  it  is  proof  against  accidents,  but  he  can  do  a  great  deal  to 
reduce  the  effects  of  accidents. 

A  ballasted  track  will  prevent  the  derailed  wheels  from  breaking 
through  the  floor.  A  floor  which  permits  this  may  cause  the 
collapse  of  the  entire  span. 

A  derailed  train  at  the  end  of  a  bridge  is  likely  to  strike  the  end 
post.  Consequently,  it  is  of  great  importance  to  provide  these 
posts  in  through  spans,  and  also  the  portals,  with  a  very  robust 
constitution. 

The  use  of  inner  and  outer  guard  rails  on  bridges,  firmly 
fastened  to  ties  embedded  in  ballast,  is  good  practice,  as  also 
inside  guard  rails,  flaring  guards  and  rerailing  frogs  on  the  bridge 


134         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

approaches.  If  these  guard  rails  are  higher  than  the  track  rails 
and  spaced  so  that  the  wheels  cannot  drop  into  the  space  between 
guard  rail  and  track  rail,  the  train  runs  practically  in  a  groove. 

Bridges  which  are  exposed  to  collisions  with  floating  objects 
deserve  particular  attention  in  their  designs.  If  such  collisions 
are  due  to  causes  other  than  ships,  as  for  example,  tree  trunks, 
blocks  of  ice,  etc.,  it  may  be  sufficient  to  raise  the  main  trusses 
above  the  floorbeams,  allowing  the  latter  to  take  the  blows,  which 
would  be  principally  delivered  in  the  direction  parallel  to  their 
length. 

Then  the  stringers  and  their  connections  may  be  so  designed 
that  they  are  self-supporting,  preventing  the  collapse  of  the  bridge 
in  spite  of  a  disabled  main  truss.*  While  the  stringers  could  be 
charged  with  this  duty  for  short  spans,  it  is  hardly  feasible  to 
resort  to  such  means  for  longer  spans. 

Drawbridges  which  are  in  great  danger  of  coming  into  collision 
with  ships  are  likely  to  have  their  bottom  chord  members  struck 
and  possibly  ruptured  so  that  a  collapse  would  appear  to  be  cer- 
tain. In  such  cases  it  should  be  the  principal  aim  of  the  designer 
to  prevent  a  rupture  of  a  bridge  member  and  this  we  believe,  at 
least  in  the  case  of  a  bottom  chord  section,  can  be  done  with  a 
high  degree  of  success  and  at  no  great  expense.  Indeed,  the 
extra  cost  should  not  play  any  role  whatever  if  all  the  evil  conse- 
quences of  a  collapse  of  a  bridge  are  considered.  The  writer 
would  design  the  stiff  web  and  bottom  chord  members  with 
the  object  of  giving  them  increased  resisting  power  against  blows, 
using  rather  solid  plates  instead  of  latticing  and  lacing,  and  paying 
of  course  great  attention  to  the  detailing  of  the  panelpoints.  As 
a  further  precaution  he  would  brace  the  bottom  chords  in  a 
horizontal  plane  against  the  track,  but  without  the  use  of  stiff 
connections.  The  track  could  be  designed  in  a  manner  that  it 
acts  like  a  cushion  to  dissipate  the  effects  of  an  impact. 

Should  the  track  be  above  the  bottom  chord,  a  collision  bracing 
may  extend  over  the  entire  width  of  the  bridge,  and  in  this  case  it 

*  See  Eng.  News,  August  10,  1906. 


EXAMPLES    AND    CONCLUDING    REMARKS  135 

would  take  at  the  same  time  the  functions  of  the  bottom  lateral 
bracing. 

If  a  bridge  has  sidewalks,  so  much  the  better.  These  can 
easily  be  constructed  to  form  an  effective  protection  for  the  main 
trusses. 

It  would  be  out  of  place  to  go  here  over  the  manifold  details 
that  are  possible.  It  suffices  to  say  that  the  conditions  which 
govern  each  individual  case  must  be  thoroughly  studied  in  order 
to  properly  solve  the  problem  at  hand. 

Before  we  conclude  we  will  say  a  few  words  in  regard  to  the 
methods  of  calculations  and  their  use. 

The  method  of  influence  lines  cannot  be  used  if  it  is  required 
to  consider  the  effects  of  deformations  of  the  bars  on  the  result, 
as  in  such  cases  the  secondary  stresses  appear  as  higher  functions 
of  the  exterior  loads.  Where  these  deformations  can  be  neglected, 
there  is  no  doubt  that  the  method  of  influence  lines  is  the  best 
that  can  be  used  in  so  far  as  it  gives  the  maximum  stresses 
required.  But  the  trouble  with  this  method  lies  in  the  fact  that 
it  involves  an  exceedingly  great  amount  of  time  and  labor,  which 
is  best  appreciated  by  the  one  who  undertakes  an  investigation  of 
this  sort.  This  must  be  the  reason  why  the  calculations  published 
so  far  are  nearly  all  made  on  the  assumption  of  only  one  position 
of  the  live  load. 

When  the  writer  undertook  the  examination  of  that  i4o-foot 
span  after  Muller-Breslau's  method,  which  is  among  the  examples, 
he  made  use  of  every  facility  he  could  think  of  to  shorten  his 
labors.  He  never  went  over  the  same  operations  the  second  time 
unless  there  was  a  necessity  to  do  so.  He  also  prepared  tables 
intended  to  facilitate  the  overlooking  of  the  situation  as  much  as 
the  nature  of  the  subject  allowed  him  to  do.  But  in  spite  of  all 
the  precautions  taken,  the  computation  proved  to  be  a  very  labori- 
ous task  indeed.  But  this  is  not  all.  It  would  be  a  mistake  to 
believe  that  that  position  of  the  live  load  which  gives  the  maxi- 
mum primary  stresses  is  at  the  same  time  the  one  corresponding 
with  the  maximum  secondary  stresses.  This  may  be  more  or 


136         SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

less  true  for  the  chords  of  a  truss  resting  on  two  supports,  but  it 
is  certainly  not  true  for  the  web  members.  The  writer  was  not 
able  to  foretell  either  the  approximate  amount  of  stress  or  the 
deformation  of  a  bar  for  a  given  position  of  the  live  load.  There 
was  only  one  way  to  find  out  something  about  these  points  and 
that  was  to  perform  the  various  operations  from  beginning  to 
end. 

The  aspect  of  this  subject  changes  considerably  in  case  we 
have  to  deal  with  the  examination  of  a  truss  for  just  one  fixed 
position  of  the  live  load.  All  methods  of  calculation  of  secondary 
stresses  require  more  time  than  those  for  primary  stresses,  but  this 
should  not  be  a  reason  to  avoid  them  in  cases  where  the  knowledge 
of  these  stresses  appears  to  be  very  desirable.  Much  depends  also 
on  the  computer  himself  in  point  of  time.  One  man  may  find  it 
a  real  hardship  to  compute  the  secondary  stresses  in  a  2oo-foot 
span  for  one  fixed  position  of  the  live  load;  while  another,  obtain- 
ing his  results  with  ease  and  rapidity,  does  not  think  much  of  it. 

There  is  no  doubt  that  cases  may  occur  where  the  safety  of  the 
structure  imposes  the  duty  on  the  designer  to  give  an  account  of 
the  secondary  stresses.  For  instance,  old  trusses  designed  for 
loads  which  are  much  lighter  than  those  they  actually  carry  are 
very  good  subjects  for  examinations,  because  it  is  here  in  particu- 
lar within  the  range  of  possibilities  that  the  stresses  are  raised  to 
a  dangerous  point.  Then  there  are  trusses  whose  examination 
appears  to  be  desirable  on  account  of  peculiarities  in  their  con- 
struction, because  they  lead  us  to  the  expectation  of  high  secondary 
stresses. 

As  far  as  we  are  aware  secondary  stresses  in  trusses  of  great 
length,  as  cantilever  trusses,  arch  ribs,  etc.,  are  not  known. 
Such  trusses  have  members  whose  stresses  are  subjected  to  a 
reversal,  which  is  aggravated  by  secondary  stresses. 

The  secondary  stresses  are  taken  into  account  by  the  common 
practice  of  lowering  the  unit  stresses,  but  this  is  a  matter  of  experi- 
ence and  hardly  feasible  by  theoretical  considerations. 

Our  knowledge  of  secondary  stresses  could  be  improved  either 


EXAMPLES    AND    CONCLUDING    REMARKS  137 

in  measuring  these  stresses  by  the  use  of  suitable  instruments,  or 
by  analytical  investigations,  or  by  both.  The  writer  suggests  that 
readers  who  take  a  particular  interest  in  this  subject  and  have 
the  time  to  do  so,  examine  trusses  and  publish  their  results,  which 
cannot  fail  to  be  instructive  as  they  would  show  us  where  we  have 
failed. 


CHAPTER   XI. 

LITERATURE. 

CONCERNING   SECONDARY   STRESSES. 

Asimont.  Hauptspannung  und  Sekundarspannung.  Zeitschrift  fiir  Baukunde, 
1880. 

Manderla.  Die  Berechnung  der  Sekundarspannungen,  welche  im  einfachen 
Fachwerke  infolge  starrer  Knotenverbindungen  entstehen.  Allgemeine  Bauzei- 
tung, 1880,  p.  34. 

Jebens.  Die  Spannungen  in  den  Vertikalstandern  der  eisernen  Briicken. 
Zeitschrift  des  Vereins  deutscher  Ingenieure,  1880,  p.  127. 

Manderla.  Formanderung  des  Fachwerks  bei  wechselnder  Belastung.  Allge- 
meine Bauzeitung,  1884,  p.  81,  89. 

Engesser.  Die  Sicherung  offener  Briicken  gegen  Ausknicken.  Centralblatt 
der  Bauverwaltung,  1884,  p.  415;  1885,  p.  71. 

Ritter.  Uber  die  Druckfestigkeit  stabformiger  Korper  mit  besonderer 
Riicksicht  auf  die  im  steifen  Fachwerk  auftretenden  Nebenspannungen.  Schwei- 
zerische  Bauzeitung,  1884,  I,  p.  37,  43,  47. 

Mliller-Breslau.  Uber  Biegungsspannungen  in  Fachwerken.  Allgemeine 
Bauzeitung,  1885,  p.  85,  89. 

Landsberg.  Ebene  Fachwerkssysteme  mit  festen  Knotenpunkten  und  das 
Princip  der  Deformationsarbeit.  Centralblatt  der  Bauverwaltung,  1885,  p.  165. 

Landsberg.  Beitrag  zur  Theorie  der  Fachwerke  (graphische  Ermittelung  der 
Sekundarspannungen  infolge  fester  Knotenverbindungen  der  Gurtstabe).  Zeit- 
schrift des  Architekten-und  Ingenieur-Vereins  zu  Hannover,  1885,  p.  361. 

Miiller-Breslau.  Beitrag  zur  Theorie  des  Fachwerks.  Zeitschrift  des  Archi- 
tekten-und Ingenieur-Vereins  zu  Hannover,  1885,  p.  417. 

Weyrauch.     Aufgaben  zur  Theorie  elastischer  Korper.     Leipzig,  1885,  p.  269. 

Manderla.  Uber  die  Wirkungsweise  gelenkformiger  Knotenverbindungen. 
Allgemeine  Bauzeitung,  1886,  p.  9,  20,  32,  37. 

Landsberg.  Beitrag  zur  Theorie  der  Fachwerke.  Zeitschrift  des  Architekten- 
und  Ingenieur-Vereins  zu  Hannover,  1886,  p.  195. 

Miiller-Breslau.  Zur  Theorie  der  Biegungsspannungen  in  Fachwerktragern. 
Zeitschrift  des  Architekten-und  Ingenieur-Vereins  zu  Hannover,  1886,  p.  399. 

Winkler.     Aussere  Krafte  gerader  Trager,  1886,  p.  166  and  1875,  p.  169,  170. 

Winkler.     Querkonstruktionen,  p.  179-182. 

Landsberg.  Beitrag  zur  Theorie  des  ebenen  Fachwerks.  Festschrift  der 
technischen  Hochschule  zu  Darmstadt,  1886. 

138 


LITERATURE  139 

Considere.  Note  sur  les  effets  anormaux  dans  les  ouvrages  metalliques. 
Annales  des  ponts  et  chaussees,  1887,  I,  p.  372. 

Frankel  und  Kniger.  Spannungs-und  Formanderungsmessungen  an  dem 
eisernen  Pendelpfeiler  Viadukte  iiber  das  Oschiitzthal  bei  Weida.  Civil  Ingenieur 
1887,  p.  439.  Nebenspanmmgen  der  Pfeiler,  p.  484. 

Allievi.  Equilibrio  internio  delle  pile  metalliche.  Roma,  1882.  (Translated 
into  German  by  Totz,  Wien,  1888.) 

Hacker.  Uber  Biegungspannungen  in  Schwedler'schen  Kuppeln.  Zeitschrift 
des  Architekten-und  Ingenieur-Vereins  zu  Hannover,  1888,  p.  223. 

Miiller-Breslau.  Beitrag  zur  Theorie  der  ebenen  elastischen  Trager.  Zeit- 
schrift des  Architekten-und  Ingenieur-Vereins  zu  Hannover,  1888,  p.  605. 

Ritter.  Anwendungen  der  graphischen  Statik.  II,  Das  Fachwerk.  Zurich, 
1890,  p.  171. 

Handbuch  der  Ingenieurwissenschaften.     Vol.  II.  1890. 

Brick.  Fachwissenschaftliche  Erorterungen  zu  dem  Berichte  des  Briicken- 
materialkomites  iiber  die  durchgefiihrten  Versuche  mit  genieteten  Tragern  aus 
Flusseisen  und  Schweisseisen.  Zeitschrift  des  Osterreichischen  Ingenieur-und 
Architekten-Vereins,  1891,  p.  76. 

Jebens.  Die  seitliche  Standsicherheit  von  eisernen  Briicken  ohne  oberen 
Querverband.  Centralblatt  der  Bauverwaltung,  1892,  p.  148. 

Engesser.  Die  seitliche  Standfestigkeit  offener  Briicken.  Centralblatt  der 
Bauverwaltung,  1892,  p.  349. 

Engesser.  Die  Zusatzkriifte  und  Nebenspannungen  eiserner  Fachwerk- 
briicken.  I,  Die  Zusatzkrafte.  Berlin,  1892.  II,  Die  Nebenspannungen.  Berlin, 
1893. 

Barkhausen.  Der  Steifrahmen  im  Wind-und  Querverbande  geschlossener 
Trogbriicken.  Zeitschrift  des  Vereins  deutscher  Ingenieure,  1892,  p.  421,  492. 

Barkhausen.  Biegungsspannungen  in  Blechen  und  Bandern  infolge  von 
einseitiger  Verlaschung  oder  von  Uberlappungsverbindungen.  Zeitschrift  des 
Vereins  deutscher  Ingenieure,  1892,  p.  553. 

Mohr.  Die  Berechnung  der  Fachwerke  mit  starren  Knotenverbindungen. 
Der  Civil  Ingenieur:  Organ  des  Sachsischen  Ingenieur-und  Architekten-Vereins, 
1892,  p.  577;  1893,  p.  67. 

Jaquier.  Note  sur  les  efforts  secondaires  qui  peuvent  se  produire  dans  les 
systemes  articules  a  attaches  rigides.  Annales  des  ponts  et  chaussees,  1893,  I,  p. 
1142. 

Engesser.  Die  zusatzlichen  Beanspruchungen  durchgehender  (kontinuir- 
licher)  Briickenkonstruktionen.  Zeitschrift  fur  Bauwesen,  1894,  p.  305. 

Engesser.  Uber  die  Verringerung  der  Nebenspannungen  von  Fachwerk- 
briicken  durch  die  Art  der  Aufstellung.  Centralblatt  der  Bauverwaltung,  1895, 

P-  3i7- 

Rapport  sur  les  epreuves  de  charge  jusqu'  a  rupture  de  1'ancien  pont  sur 
1'Emme  a  Wolhusen.  Berne,  1895. 

Haseler.  Berechnung  der  auf  Verdrehung  beanspruchten  Briickentrager.  Zeit- 
schrift des  Vereins  deutscher  Ingenieure,  1896,  p.  761. 


140        SECONDARY    STRESSES    IN    BRIDGE    TRUSSES 

Dupuy.  Resistances  des  barres  soumises  a  des  efforts  agissant  parallelement  & 
leur  axe  neutre  et  en  dehors  de  cette  axe.  Annales  des  ponts  et  chausse'es,  1896, 
II,  p.  223. 

Haseler.  Der  Briickenbau.  I,  Die  eisernen  Briicken.  3.  Lief.  Braunschweig, . 
1897. 

Luegers.  Lexikon  der  gesammten  Technik  mit  ihren  Hiilfswissenschaften  im 
Verein  mit  Fachgenossen  herausgegeben. 

Franke.  Berechnung  der  Durchbiegung  und  der  Nebenspannungen  der 
Fachwerktrager.  Zeitschrift  fiir  Bauwesen,  1898. 

Patton.  Beitrag  zur  Berechnung  der  Nebenspannungen  infolge  starrer 
Knotenverbindungen  bei  Briickentragern.  Zeitschrift  fiir  Architektur-and 
Ingenieurwesen.  Heft  4,  1902. 

Isami  Hiroi.     Statically  Indeterminate  Bridge  Stresses.     1905. 

Mehrtens.  Vorlesungen  iiber  Statik  der  Baukonstruktionen  und  Festig- 
keitslehre.  Vol.  III.  1905. 

Mohr.     Abhandlungen  aus  dem  Gebicte  der  technischen  Mechanik.     1906. 

CONCERNING  IMPACT  AND    VIBRATIONS. 

Resal.  Effet  des  charges  roulantes.  Annales  des  ponts  et  chaussees,  1882, 
II,  p.  337-352' 

Resal.  Effet  des  charges  roulantes  sur  les  ponts  metalliques.  Annales  des 
ponts  et  chaussees,  1883,  I,  p.  277-299. 

Robinson.  Vibration  of  bridges.  Transactions  of  the  American  Society  of 
Civil  Engneers,  1887,  Vol.  XVI. 

Soulyere.  Action  dynamiques  des  charges  roulantes  sur  les  poutres  rigides 
qui  ne  travaillent  qu'  a  la  flexion.  Annales  des  ponts  et  chaussees,  1889, 
p.  341-441. 

Glauser.  Dynamische  Wirkungen  bewegter  Lasten  auf  eiserne  Briicken. 
Glaser's  Annalen  fiir  Gewerbe  und  Bauwesen,  1891,  Vol.  29,  p.  113;  1892,  Vol. 
30,  p.  61;  1894,  Vol.  34,  p.  56. 

Zimmermann.  Die  Wirkungen  bewegter  Lasten  auf  Briicken.  Centralblatt 
der  Bauverwaltung,  1891,  p.  448;  1892,  p.  159,  199,  215. 

Melan.  Uber  die  dynamische  Wirkung  bewegter  Lasten  auf  Briicken.  Zeit- 
schrift des  osterreichischen  Ingenieur-und  Architekten-Vereins,  1893,  p.  293. 

Melan.  Uber  die  dynamische  Wirkung  bewegter  Lasten  auf  eiserne  Briicken. 
Glaser's  Annalen  fur  Gewerbe  und  Bauwesen,  1894. 

Deslandres.  Note  sur  les  epreuves  par  charge  roulante  et  1'action  des  chocs. 
Annales  des  ponts  et  chaussees,  1894,  I,  p.  735. 

Zimmermann.  Die  Schwingungen  eines  Tragers  mit  bewegter  Last.  Berlin, 
1896. 

Stone.  The  determination  of  the  safe  working  stress  for  railway  bridges  of 
wrought  iron  and  steel.  Transactions  of  the'American  Society  of  Civil  Engineers, 
1889.  Vol.  XLI. 

Turneaure.  Some  experiments  on  bridges  under  moving  train-loads.  Trans- 
actions of  the  American  Society  of  Civil  Engineers,  1899,  Vol.  XLI. 


SHORT-TITLE     CATALOGUE 

OF  THE 

PUBLICATIONS 

OF 

JOHN   WILEY   &    SONS, 

NEW  YORK. 
LONDON:   CHAPMAN  &  HALL,  LIMITED. 


ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application.      Books  marked  \vith    an  asterisk  (*)  are  sold 
at    net  prices   only.       All  books  are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

Armsby's  Manual  of  Cattle-feeding i2mo,  Si  75 

Principles  of  Animal  Nutrition 8vo,  4  oo 

Budd  and  Hansen's  American  Horticultural  Manual: 

Part  I.  Propagation,  Culture,  and  Improvement i2mo,  i  50 

Part  II.  Systematic  Pomology 121110,  i  50 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

Graves's  Forest  Mensuration ' 8vo,  4  oo 

Green's  Principles  of  American  Forestry i2mo,  i  50 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,  2  oo 

Hanausek's  Microscopy  of  Technical  Products.     (Winton.) 8vo,  5  oo 

Herrick's  Denatured  or  Industrial  Alcohol 8vo,  4  oo 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration i2mo,  i  50 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i  50 

Sanderson's  Insects  Injurious  to  Staple  Crops i2mo,  i  50 

*Schwarz's  Longleaf  Pine  in  Virgin  Forest 121110,  i  25 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

WolFs  Handbook  for  Farmers  and  Dairymen i6mo,  i  50 


ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Bashore's  Sanitation  of  a  Country  House I2mo.  i  oo 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Birkmire's  Planning  and  Construction  of  American  Theatres 8vo,  3  oo 

Architectural  Iron  and  Steel 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  oo 

Planning  and  Construction  of  High  Office  Buildings 8vo,  3  50 

Skeleton  Construction  in  Buildings 8^0,  3  oo 

Brigg's  Modern  American  School  Buildings 8vo,  4  oo 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  oo 

1 


Freitag's  Architectural  Engineering 8vo.  3  50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Sanitation  of  Public  Buildings .  I2mo,  i  50 

Theatre  Fires  and  Panics I2mo,  i  50 

*Greene's  Structural  Mechanics 8vo,  2  50 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,  75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Kellaway 's  How  to  Lay  Out  Suburban  Home  Grounds 8vo,  2  oo 

Kidder's  Architects' and  Builders' Pocket-book.  Rewritten  Edition.  i6mo,mor.,  5  oo 

Merrill's  Stones  for  Building  and  Decoration .  .8vo,  5  oo 

Non-metallic  Minerals:    Their  Occurrence  and  Uses 8vo,  4  oo 

Monckton's  Stair-building 4to,  4  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Peabody's  Naval  Architecture 8vo,  7  50 

Rice's  Concrete-block  Manufacture 8vo,  2  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  oo 

*  Building  Mechanics'  Ready  Reference  Book: 

*  Carpenters'  and  Woodworkers'  Edition i6mo,  morocco,  i  50 

*  Cementworkers  and  Plasterer's  Edition.     (In  Press.) 

*  Stone-  and  Brick-mason's  Edition i2mo,  mor.,  i  50- 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  oo 

Towne's  Locks  and  Builders'  Hardware i8mo,  morocco,  3  oo 

Turneaure  and  Maurer's  Principles  of  Reinforced  Concrete  Construc- 
tion   8vo,  3  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Wilson's  Air  Conditioning,     ijn  Press.) 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,  4  oo 
Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

I2mo,  i  25 

The  World's  Columbian  Exposition  of  1893 Large  4to,  i  oo 


ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50° 

Chase's  Screw  Propellers  and  Marine  Propulsion 8vo,  3  oo 

Cloke's  Gunner's  Examiner 8vo,  i  50 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

Sheep,  7  50 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 24mo,  morocco,  2  oo 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dudley's  Military  Law  and  the  Procedure  of  Courts-martial. .  .  Large  i2mo,  2  50 
Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  oo 

2 


*  Dyer's  Handbook  of  Light  Artillery i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

*  Fiebeger's  Text-book  on  Field  Fortification Small  8vo,  2  oo 

Hamilton's  The  Gunner's  Catechism i8mo,  i  oo 

*  Hoff' s  Elementary  Naval  Tactics 8vo,  I  50 

Ingalls's  Handbook  of  Problems  in  Direct  Fire 8vo,  4  oo 

*Lissak's  Ordnance  and  Gunnery 8vo,  6  oo 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  .8vo,  each,  6  oo 

*  Mahan's  Permanent  Fortifications.     (Mercur.) 8vo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  I  50 

*  Mercur's  Attack  of  Fortified  Places i2mo,  2  oo 

*  Elements  of  the  Art  of  War 8vo,  4  oo 

Metcalf's  Cost  of  Manufactures — And  the  Administration  of  Workshops.  .8vo,  5  oo 

*  Ordnance  and  Gunnery.     2  vols I2mo,  5  oo 

Murray's  Infantry  Drill  Regulations. i8mo,  paper,  10 

Nixon's  Adjutants'  Manual 241110,  i  oo 

Peabody's  Naval  Architecture 8vo,  7  50 

*  Phelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner i2mo,  4  oo» 

Sharpe's  Art  of  Subsisting  Armies  in  War i8mo,  morocco,  i  5o> 

*  Tupes  and  Poole's  Manual  of  Bayonet  Exercises  and    Musketry  Fencing. 

24010,  leather,  so- 
Weaver's  Military  Explosives 8vo,  3  oo 

Wheeler's  Siege  Operations  and  Military  Mining 8vo,  2  oo 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50. 

Young's  Simple  Elements  of  Navigation i6mo,  morocco,  2  oo> 

ASSAYING. 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  .  .  .8vo,  3  oo 

Low's  Technical  Methods  of  Ore  Analysis 8vo,  3  oo 

Miller's  Manual  of  Assaying i2mo,  i  oo 

Cyanide  Process i2mo,  i  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) i2mo,  2  50 

O'DriscolI's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process T I2mo,  i  50 


ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Craig's  Azimuth 4to,  3  50 

Crandall's  Text-book  on  Geodesy  and  Least  Squares 8vo,  3  oo 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Merrirran's  Elements  or  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  oo 

3 


BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

1 6mo,  morocco,  i  25 

Thome  and  Bennett's  Structural  and  Physiological  Botany i6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo,  2  oo 

CHEMISTRY. 

*  Abegg's  Theory  of  Electrolytic  Dissociation.    (Von  Ende.) i2mo,  i  25 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables i2mo,  i  25 

Alexeyeff's  General  Principles  of  Organic  Synthesis.     (Matthews.) 8vo,  3  oo 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Beard '  s  Mine  Gases  and  Explosions .     ( I  n  P  r ess . ) 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolduan's  Immune  Sera 12mo ,  i  50 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  i  50 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy Svo,  4  oo 

*  Claassen's  Beet-sugar  Manufacture.     (Hall  and  Rolfe.) 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood.).  .8vo,  3  oo 

Cohn's  Indicators  and  Test-papers i2mo,  2  oo 

Tests  and  Reagents 8vo,  3  oo 

Crafts's  Short  Course  in  Qualitative  Chemical  Analysis.   (Schaeffer.).  .  .  i2mo,  i  50 

*  Danneel's  Electrochemistry.     (Merriam.) lamo,  i   25 

Dolezalek's   Theory   of   the   Lead   Accumulator   (Storage   Battery).         (Von 

Ende,) I2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) I2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) Svo,  4  oo 

Eissler's  Modern  High  Explosives Svo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) Svo,  3  oo 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) I2mo,  i   25 

*  Fischer's  Physiology  of  Alimentation Large  I2mo,  2  oo 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

I2mo,  morocco,  i  50 

Fowler's  Sewage  Works  Analyses I2mo,  2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) Svo,  5  oo 

Manual  of  Qualitative  Chemical  Analysis.  Part  I.  Descriptive.  (Wells.)  Svo,  3  oo 

Quantitative  Chemical  Analysis.     (Cohn.)     2  vols Svo,  12  50 

Fuertes's  Water  and  Public  Health I2mo,  i  50 

Furman's  Manual  of  Practical  Assaying Svo,  3  oo 

*  Getman's  Exercises  in  Physical  Chemistry I2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers. I2mo,  i  25 

*  Gooch  and  Browning's  Outlines  of  Qualitative  Chemical  Analysis.  Small  Svo,  i  25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,  2  oo 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall) i2mo,  i  25 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) Svo,  4  oo 

Hanausek'  s  Microscopy  of  Technical  Products.     (Winton. ) svo,  5  oo 

*  Haskin's  and  MacLeod's  Organic  Chemistry 12mo,  2  oo 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo,  i  50 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Herrick's  Denatured  or  Industrial  Alcohol Svo,  4  oo 

Hind's  Inorganic  Chemistry Svo,  3  oo 

*  Laboratory  Manual  for  Students I2tno,  i  oo 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) Svo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) Svo,  2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) X2mo,  I  oo 

4 


3  oo 
5  oo 
i  25 


2  50 

1  OO 

3  oo 

2  00 


7  50 
3  oo 
3  oo 
3  oo 

I  OO 

I  50 


Eolley  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments ,  and  Varnishes. 
(In  Press) 

Hopkins's  Oil-chemists'  Handbook 8vo, 

Iddings's  Rock  Minerals 8vo, 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo, 
Johannsen's  Key  for  the  Determination  of  Rock -forming  Minerals  in  Thin  Sec- 
tions.    (In  Press) 

Xeep's  Cast  Iron 8vo, 

Xadd's  Manual  of  Quantitative  Chemical  Analysis i2mo, 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo, 

*  Langworthy  and  Austen.        The   Occurrence   of  Aluminium  in  Vegetable 

Products,  Animal  Products,  and  Natural  Waters 8vo, 

iassar-Cohn's  Application  of  Some  General  Reactions  to  Investigations  in 

Organic  Chemistry.  (Tingle.) i2mo, 

JLeach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo, 

Lob's  Electrochemistry  of  Organic  Compounds.  (Lorenz.) 8vo, 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  ..  ,8vo, 

Low's  Technical  Method  of  Ore  Analysis 8vo, 

Lunge's  Techno-chemical  Analysis.  (Cohn.) i2mo 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo, 

Maire's  Modern  Pigments  and  their  vehicles.     (In  Press.) 

Mandel's  Handbook  for  Bio-chemical  Laboratory I2mo, 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  I2mo, 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo, 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo, 

Matthew's  The  Textile  Fibres,     ad  Edition,  Rewritten 8vo", 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .i2mo, 

Miller's  Manual  of  Assaying i2mo, 

Cyanide  Process I2mo, 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.).  .  .  .  i2mo, 

Mixter's  Elementary  Text-book  of  Chemistry I2mo, 

Morgan's  An  Outline  of  the  Theory  of  Solutions  and  its  Results i2mo, 

Elements  of  Physical  Chemistry i2mo, 

*  Physical  Chemistry  for  Electrical  Engineers i2mo, 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco, 

*  Muir's  History  of  Chemical  Theories  and  Laws 8vo, 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo, 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo, 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) i2mo, 

"  "  "  Part  Two.     (Turnbull.) i2mo, 

*  Palmer's  Practical  Test  Book  of  Chemistry 12mo, 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.     (Fischer.) ....  I2mo, 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper, 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo, 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo, 

Poole's  Calorific  Power  of  Fuels 8vo, 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo, 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Standpoint..8vo ,    2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo, 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo, 

Disinfection  and  the  Preservation  of  Food 8vo, 

Riggs's  Elementary  Manual  for  the  Chemical  Laboratory 8vo, 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

5 


i  So 
60 

4  oo 
i  25 
4  oo 

00 

oo 
oo 

50 
50 

00 

3  oo 

5  oo 

1  50 

4  oo 

5  oo 

2  oo 

1  50 

2  00 
1  00 

i  25 

50 
5  oo 

I  50 

3  00 

i  25 


3  oo 

4  oo 
4  oo 
i  25 
4  oo 


Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo> 

*  Whys  in  Pharmacy i2mo,  i  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish .  .8vOf  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.). .  .  .  .8vo,  2  50 

Schimpf's  Text-book  of  Volumetric  Analysis i2mo,  2  50 

Essentials  of  Volumetric  Analysis ,  i2mo,  i  25 

*  Qualitative  Chemical  Analysis 8vo,  i    25 

Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students .8vo,  2  50' 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo.  morocco,  3  oo 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

*  Descriptive  General  Chemistry .8vo»  3  oo- 

Treadwell's  Qualitative  Analysis.     (Hall.) • 8vo,  3  oo 

Quantitative  Analysis.     (Hall.) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Beet-sugar  Manufacture  and  Refining.     Vol.  I Small  8vo,  4  oo 

Vol.11 Small  8vo,  500 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oo 

Weaver's  Military  Explosives 8vo,  3  oo 

Wehrenfennig's  Analysis  and  Softening  of  Boiler  Feed-Water 8vo,  4  oo 

Wells's  Laboratory  Guide  in*Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students. .  .i2mo,  i  50 

Text-book  of  Chemical  Arithmetic I2mo,  i  25 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process I2mo,  i  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry i2mo,  2  oo 


CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS    OF    ENGINEERING 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments i2mo,  3  oo 

Bixby's  Graphical  Computing  Table Paper  io£  X  24^  inches.  25 

Breed  and  Hosmer's  Principles  and  Practice  of  Surveying 8vo,  3  oo 

*  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal 8vo,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

*  CorthelTs  Allowable  Pressures  on  Deep  Foundations izmo,  125 

Crandall's  Text-book  on  Geodesy  and  Least  Squares 8vo,  3  oo 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  r  oo 

*Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  oo 

Flemer's  Phototopographic  Methods  and  Instruments 8vo,  5  oo 

Folwell's  Sewerage.      (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  5<> 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements i2mo,  i  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

*  Hauch  and  Rice 's  Tables  of  Quantities  for  Preliminary  Estimates I2mo,  i  25 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 


Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Howe's  Retaining  Walls  for  Earth 12010,  i  25 

Hoyt  and  Grover's  River  Discharge 8vo,  2  oo 

*  Ives's  Adjustments  of  the  Engineer's  Transit  and  Level i6mo,  Bds.  25 

Ives  and  Hilts's  Problems  in  Surveying i6mo,  morocco,  i  50 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.).  12010,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

*  Descriptive  Geometry.    8vo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying 8vo,  3  50 

Ogden's  Sewer  Design i2mo,  2  oo 

Parsons's  Disposal  of  Municipal  Refuse 8vo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  4  oo 

Riemer's  Shaft-sinking  under  Difficult  Conditions.     (Corning  and  Peele.) .  .8vo,  3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

•Tracy's  Plane  Surveying I6mo,  morocco,  3  oo 

•*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Venable's  Garbage  Crematories  in  America 8vo,  2  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i   25 

Wilson's  Topographic  Surveying 8vo,  3  50 

BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations 8vo,  3  oo 

Design  and  Construction  of  Metallic  Bridges 8vo,  5  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II JTirall  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

<Greene's  Roof  Trusses 8vo,  i   25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Grimm's  Secondary  Stresses  in  Bridge  Trusses.     (In  Press. ) 

Howe's  Treatise  on  Arches 8vo,  4  oo 

Design  of  Simple  Roof -trusses  in  Wood  and  Steel 8vo,  2  oo 

Symmetrical  Masonry  Arches 8vo,  2  50 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.    Stresses  in  Simple  Trusses 8vo,  2  50 

Part  II.    Graphic  Statics 8vo,  2  50 

Part  III.  Bridge  Design 8vo,  2  50 

Part  IV.  Higher  Structures 8vo,  2  50 

7 


Morison's  Memphis  Bridge.     4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,  2  oo 

Specifications  for  Steel  Bridges i2mo,  50- 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume 8vo,  3  50 

HYDRAULICS. 

Barnes's  Ice  Formation 8vo,  3  oo> 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  oa 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Hydraulic  Motors 8vo,  2  oo 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power.  . i2mo,  3  oo 

Folwell's  Water-supply  Engineering 8vo,  4  oo 

Frizell's  Water-power , 8vo,  5  oo 

Fuertes's  Water  and  Public  Health i2mo,  i   50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Clean  Water  and  How  to  Get  It Large  I2mo,  1  5o 

Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water- works : 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

*  Hubbard  and  Kiersted's  Water- works  Management  and  Maintenance..  .8vo,  4  ca 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics. 8vo,  4  oa 

Schuyler's   Reservoirs   for   Irrigation,    Water-power,   and    Domestic   Water- 
supply Large  8vo,  5  oo> 

*  Thomas  and  Watt's  Improvement  of  Rivers 4to,  6  oa 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams,     sth  Edition,  enlarged.  .  .  4to,  6  oa 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 .  .4to,  10  oa 

Whipple's  Value  of  Pure  Water Large  i2mo,  i  oo- 

Williams  and  Hazen's  Hydraulic  Tables 8vo,  i  50 

Wilson's  Irrigation  Engineering , Small  8vo,  4  oo- 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo> 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo,  5  oo 

Roads  and  Pavements 8vo,  5  oo 

Black's  United  States  Public  Works Oblong  4to,  5  oo- 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Byrne's  Highway  Construction 8vo,  5  oo- 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo« 

Church's  Mechanics  of  Engineering 8vo,  6  oo>- 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  410.  7  5<> 

*Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 


Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Graves's  Forest  Mensuration 8vo,  4  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  SO 

Martens's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  SO 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  rnor.,  4  oo 

*  Ries's  Clays:  Their  Occurrence,  Properties,  and  Uses 8vo,  5  oo 

Rockwell's  Roads  and  Pavements  in  France I2mo,  i   25 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  ard  Varnish 8vo,  3  oo 

*Schwarz's  Longleaf  Pine  in  Virgin  Forest  ...   i2tno,  i    25 

Smith's  Materials  of  Machines i2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Text-book  on  Roads  and  Pavements i2mo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  .  5  oo 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  oo 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Turneaure  and  Maurer's  Principles  of  Reinforced  Concrete  Construction.     8vo,  3  oo 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.).  .  i6mo,  mor.,  2  oo 

*  Specifications  for  Steel  Bridges i2mo,  50 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and 

Steel.  ...                8vo,  4  oo 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Crockett's  Methods  for  Earthwork  Computations.     (In  Press) 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book .  .  i6mo,  morocco  5  oo 

Dredge's  History  of  the  Pennsylvania  Railroad:    (1879) Paper,  5  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .  i6mo,  mor.,  2  50 
Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Raymond's  Elements  of  Railroad  Engineering.     (In  Press.) 

9 


Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral. i6mo,  morocco,  x  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  x  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo 

Economics  of  Railroad  Construction Large  i2mo,  2  50 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  Svo,  5  oo 


DRAWING.     • 

Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bartlett's  Mechanical  Drawing Svo ,  3  oo 

*  "  "  "       Abridged  Ed Svo,  150 

Coolidge's  Manual  of  Drawing Svo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers   .  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines Svo,  4  oo 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications Svo,  2  50 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective Svo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing Svo,  2  50 

Advanced  Mechanical  Drawing Svo,  2  oo 

Jones's  Machine  Design : 

Part  I.     Kinematics  of  Machinery Svo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts Svo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry Svo,  3  oo 

Kinematics;  or,  Practical  Mechanism Svo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams Svo,  i  50 

MacLeod's  Descriptive  Geometry Small  Svo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting Svo,  i  50 

Industrial  Drawing.  (Thompson.) Svo,  3  50 

Moyer's  Descriptive  Geometry Svo,  2  oo 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing Svo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo,  3  oo 

Robinson's  Principles  of  Mechanism Svo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo,  3  oo 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.  (McMillan.) Svo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design Svo,  3  oo 

*  Titsworth's  Elements  of  Mechanical  Drawing Oblong  Svo,  i   25 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo,  i  oo 

Drafting  Instruments  and  Operations i2mo,  i  25 

Manual  of  Elementary  Projection  Drawing i2mo,  i   50 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo,  r  oo 

Plane  Problems  in  Elementary  Geometry i2mo,  125 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective Svo,  3  50 

General  Problems  of  Shades  and  Shadows Svo,  3  oo 

Elements  of  Machine  Construction  and  Drawing Svo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry Svo,  2  50 

Weisbach's    Kinematics    and    Power    of    Transmission.        (Hermann    and 

Klein.) » Svo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo.  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying Svo,  3  50 

10 


Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  i  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  oo 

ELECTRICITY  AND  PHYSICS. 

*  Abegg's  Theory  of  Electrolytic  Dissociation.     (Von  Ende.) i2mo,  i   25 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  i2mo,  i  oo 

Benjamin's  History  of  Electricity 8vo,  3  oo 

Voltaic  Cell 8vo,  3  oo 

Betts's  Lead  Refining  and  Electrolysis.     (In  Press.) 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo,  3  oo 

*  Collins's  Manual  of  Wireless  Telegraphy i2tno,  i  50 

Morocco,  2  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

*  Danneel's  Electrochemistry.     (Merriam.) i2mo,  i  25 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's  Theory  of  the  Lead  Accumulator  (Storage  Battery).    (Von  Ende.) 

i2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  co 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  2  50 

Hanchett's  Alternating  Currents  Explained i2mo,  i  oo 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Hobart  and  Ellis's  High-speed  Dynamo  Electric  Machinery.     (In  Press.) 

Holman's  Precision  of  Measurements 8vo,  2  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and   Tests.  .  .  .Large  8vo,         75 
Karapetoff's  Experimental  Electrical  Engineering.     (In  Press.) 

Kinzbrunner's  Testing  of  Continuous-current  Machines 8vo,  2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard— Burgess.)  i2mo,  3  oo 

Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo,  3  oo 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  8vo,  each,  6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,  2  50 

Norris's  Introduction  to  the  Study  of  Electrical  Engineering.     (In  Press.) 

*  Parshall  and  Hobart's  Electric  Machine  Design 4to,  half  morocco,  12  50 

Reagan's  Locomotives:    Simple,  Compound,  and  Electric.      New  Edition. 

Large  12 mo,  3  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo, 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo, 

Thurston's  Stationary  Steam-engines ' 8vo, 

*  Tillman's  Elementary  Lessons  in  Heat.  .  .  .  .8vo, 


Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo, 


oo 
50 
50 
50 
oo 
Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,    2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,    7  oo 

*  Sheep,     7  50 

*  Dudley's  Military  Law  and  the  Procedure  of  Courts-martial  .  .  .    Larpe  i2mo,     2  50 

Manual  for  Courts-martial i6mo,  morocco,     i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Winthrop's  Abridgment  of  Military  Law x  21110,  2  50 

11 


Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving I2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers lamo,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing .8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission , .  8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.    (Pope,  Haven,  and  Dean.)  .  .  8vo,  4  oo 
MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

MacFarland's  Standard  Reduction  Factors  for  Gases 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing ' 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air i2mo,  i   50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design. 8vo,  3  oo 

Thurston's    Treatise    on    Friction  and    Lost    Work   in    Machinery   and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  i2mo,  i  oo 

Tillson's  Complete  Automobile  Instructor i6mo,  i  50 

Morocco,  2  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

MATERIALS   OF   ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition. 

Reset 8vo,  7  5<> 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

14 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,    2  oo 

Elements  of  Analytical  Mechanics 8vo,    3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,    4  oo 

STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram i2mo,    i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,     i  50 

Creighton's  Steam-engine  and  other  Heat-motors         8vo,    500 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  . .  .i6mo,  mor.,    5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,     i  oo 

Goss's  Locomotive  Sparks 8vo,    2  oo 

Locomotive  Performance 8vo,    5  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,    2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,    5  oo 

Heat  and  Heat-engines 8vo      5  oo 

Kent's  Steam  boiler  Economy 8vo,    4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,     i  50 

MacCord's  Slide-valves 8vo,    2  oo 

Meyer's  Modern  Locomotive  Construction .- 4 to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator 12 mo,     T  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors   8vo,     i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,    5  oo 

Valve-gears  for  Steam-engines 8vo,    2  50 

Peabody  and  Miller's  Steam-boilers 8vo,    4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,    2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,     i  25 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric.     New  Edition. 

Large  i2mo,    3  50 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,    2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,    2  50 

Snow's  Steam-boiler  Practice 8vo,    3  oo 

Spangler's  Valve-gears 8vo,    2  50 

Notes  on  Thermodynamics i2mo,     i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,    3  oo 

Thomas's  Steam-turbines 8vo,    3  50 

Thurston's  Handy  Tables 8vo,    i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory. 8vo,    6  oo 

Part  II.     Design,  Construction,  and  Operation 8vo,    6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,    5  oo 

Stationary  Steam-engines 8vo,    2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,    i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation. 8vo,    5  oo 
Wehrenfenning's  Analysis  and  Softening  of  Boiler  Feed-water  (Patterson)  8vo,     4  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,    5  oo 

Whitham's  Steam-engine  Design 8vo,    5  oo 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  ..8vo,    4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8«ro,  2  50 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures   8vo,  7  50 

Chase's  The  Art  of  Pattern-making I2mo,  2  50 

15 


Church's'  Mechanics  of  Engineering 8vo,  6  oo 

Notes  and  Examples  in  Mechanics 8vo,  2  oo 

Compton's  First  Lessons  in  Metal- working izmo,  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  50 

Treatise  on  Belts  and  Pulleys I2mo,  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .i2mo,  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  oo 

Dredge's   Record  of   the   Transportation  Exhibits   Building  of   the   World's 

Columbian  Exposition  of  1893 4to  half  morocco,  3  oo 

Du  Bois's  Elementary  Principles  of  Mechanics : 

Vol.      I.     Kinematics 8vof  3  50 

Vol.    II.     Statics 8vo,  4  oo 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Locomotive  Performance 8vo,  5  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  i  oo 

Hobart  and  Ellis 's  High-speed  Dynamo  Electric  Machinery.     (In  Press.) 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i   50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)  .8vo,  4  oo 
MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

*  Martin's  Text  Book  on  Mechanics,  Vol.  I,  Statics i2mo,  i   25 

*  Vol.  2,  Kinematics  and  Kinetics  .  .I2mo,  1  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Elements  of  Mechanics I2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

*  Parshall  and  Hobart's  Electric  Machine  Design 4to,  hah*  morocco,  12  50 

Reagan's  Locomotives :  Simple,  Compound,  and  Electric.     New  Edition. 

Large  12010,  3  5o 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.  Vol.  1 8vo,  2  50 

Sanborn's  Mechanics :  Problems Large  i2mo,  i  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management 12010,  2  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Sorer  s  Carbureting  and  Combustion  of  Alcohol  Engines.  (Woodward  and 

Preston.) Large  8vo,  3  oo 

16 


Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo.  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  1 2mo,  i  oo 

Tillson's  Complete  Automobile  Instructor i6mo,  i   50 

Morocco,  2  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  5» 

Weisbach's  Kinematics  and  Power  of  Transmission.    (Herrmann — Klein. ).8vo.  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.). 8vo.  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics I2mo,  i  25 

Turbines. 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  I  oo 

MEDICAL. 

*  Bolduan's  Immune  Sera , 12mo,  1  50 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.).    ..  .Large  i2mo,  2  50 

Ehrlich's  Collected  Studies  on  Immunity.     (Bolduan.) 8vo,  6  oo 

*  Fischer's  Physiology  of  Alimentation Large  12mo,  cloth,  2  oo 

Hammarsten's  Text-book  on  Physiological  Chemistry.     (Mandel.) 8vo,  4  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.     (Lorenz. ) i2mo,  oo 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.     (Fischer.) ....  i2mo,  25 

*  Pozzi-Escot's  The  Toxins  and  Venoms  and  their  Antibodies.     (Cohn.).  i2mo,  oo 

Rostoski's  Serum  Diagnosis.     (Bolduan.) 12010,  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) 8vo,  50 

*  Satterlee's  Outlines  of  Human  Embryology i2mo,  25 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  50 

Von  Behrtng's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  oo 

Woodhull's  Notes  on  Military  Hygiene i6mo,  50 

*  Personal  Hygiene i2mo,  oo 

Wulling's  An  Elementary  Course  in  Inorganic  Pharmaceutical  and  Medical 

Chemistry I2mo,  2  oo 

METALLURGY. 

Betts's  Lead  Refining  by  Electrolysis.    (In  Press.) 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

Vol.  II.     Gold  and  Mercury 8vo,  7  50 

Goesel's  Minerals  and  Metals:     A  Reference  Book , .  . .  .  i6mo,  mor.  3  oo 

*  Iles's  Lead-smelting i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard— Burgess. )i2mo,  3  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users.  .  , 12010,  2  oo 

Miller's  Cyanide  Process I2mo,  i  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). . .  .  i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  eo 

Part    II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,    2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,    3  oo 

17 


Boyd's  Map  of  Southwest  Virignia Pocket-book  form,  a  oo 

*  Browning's  Introduction  to  the  Rarer  Elements Kvo,  i  50 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "System  of  Mineralogy." Large  8vo,  i  oo 

Texi-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them  ....    I2mo,  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  oo 

Manual  of  Mineralogy  and  Petrography i2mo  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  oo 

Eakle's  Mineral  Tables 8vo,  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Goesel's  Minerals  and  Metals:     A  Reference  Book ibmo,  mor.  300 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall) I2mo,  i  25 

Iddings's  Rock  Minerals 8vo,  5  oo 

Johannsen's  Key  for  the  Determination  of  Rock-forming  Minerals   in   Thin 
Sections.     (In  Press.) 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe.  I2tno,  60 
Merrill's  Non-metallic  Minerals.  Their  Occurrence  and  Uses 8vo,  4  oo 

Stones  for  Building  and  Decoration   ..                 8vo,  500 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 

Tables  of  Minerals 8vo,  l  00 

*  Richards's  Synopsis  of  Mineral  Characters i2mo.  morocco,  i   25 

*  Ries's  Clays.  Their  Occurrence.  Properties,  and  Uses 8vo,  5  oo 

Rosenbusch's    Microscopical   Physiography    of   the    Rock-making  Minerals. 

(Iddings.) 8vo,  5  oo 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo,  2  oo 

MINING. 

Beard's  Mine  Gases  and  Explosions.     (In  Press.) 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virginia Pocket-book  form,  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  I  oo 

Eissler's  Modern  High  Explosives 870.  4  oo 

Goesel's  Minerals  and  Metals :     A  Reference  Book i6mo,  mor.  3  oo 

Goodyear's  Coal-mines  of  the  Western  Coart  of  the  United  States i2mo,  2  50 

I hlseng's  Manual  of  Mining.   8vo,  5  oo 

*  Iles's  Lead-smelting i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Miller's  Cyanide  Process i2mo,  i  oo 

O'Dnscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Weaver's  Military  Explosives 8vo,  3  oo 

Wilson's  Cyanide  Processes i2mo.  i  50 

Chlorination  Process.  .  - i^mo,  i  50 

Hydraulic  and  Placer  Mining.     2d  edition,  rewritten 12 mo,  2  50 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation T2mo,  i  25 

SANITARY  SCIENCE. 

Bashore's  Sanitation  of  a  Country  House i2mo,  i  oo 

*  Outlines  ot  Practical  Sanitation i2mo,  i  25 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oo 

Water-supply  Engineering 8vo,  4  O<K 

18 


Fowler's  Sewage  Works  Analyses izmD,  2  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works 12010,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo.  i  oo 

Sanitation  of  Public  Buildings 12mo,  1  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint;  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2tno,  z  25 

*  Merriman's  Elements  of  Sanitary  Engineering 8vorf  2  oo 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

*  Price's  Handbook  on  Sanitation I2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries I2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Cost  of  Shelter i2mo,  i  oo 

Richards  and  Woodman's  Air-  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  S.wage  and  Bacterial  Purification  of  Sewage 8vo,  4  oo 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Air  Conditioning.    (In  Press.) 

Winton's  Microscopy  of  Vegetable  Foods 8/0,  7  50 

Woodhull's  Notes  on  Military  Hygiene iCmo,  i  50 

*  Personal  Hygiene i2mo,  i  oo 

MISCELLANEOUS. 

Association  of   State   and  National  Food  and  Dairy  Departments  (Interstate 
Pure  Food  Commission; : 

Tenth  Annual  Convention  Held  at  Hartford,  July   17-20,  1906.  ..  8vo,  3  oo 
Eleventh   Annual    Convention.    Held  at  Jamestown   Tri -Centennial 

Exposition,  July  16-19,  1907.     (In  Press.) 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  Cvo,  i  50 

Fen-el's  Popular  Treatise  on  the  Winds 8vo,  4  oo 

Gannett's  Statistical  Abstract  of  the  World    24010,  75 

Gerhard's  The  Modern  Bath  and  Bath-houses.     (In  Press  ) 

Haines's  American  Railway  Management I2mo,  2  50 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  .Small  8vo,  3  oo 

Rotherham's  Emphasized  New  Testament. Large  8vo,  2  o<> 

Standage's  Decorative  Treatment  of  Wood,  Glass,  Metal,  etc.     (In  Press.) 

The  World's  Columbian  Exposition  of  1803 4to,  i  oo 

Winslow's  Elements  of  Applied  Microscopy I2mo,  i  50 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar I2mo,     i  25 

Hebrew  Chrestomathy 8vo,    2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco      5  oo 

Letteris's  Hebrew  Bible 8vo,    2  25 

19 


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UNIVERSITY    Ur 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


NOV  13 


UWARYUW 

APR28 


REC'D 

APR  2  8  1962 


RSC'D  LD 

MAY  1  2  1962 


laou 


30M-V15 


